At least as old as The Bible itself, as shown by the case of Sarah, daughter of Raguel, saved by Tobias with Raphael's help.
Played straight in numerous medieval tales from all over Europe, with Lyonesse, Guinevere and Iseult as model examples. In the Chivalric Romance, it was particularly noted as an element of the Matter of Britain, which was the supreme matter dealing with love.
Mercilessly subverted way back in 1495 in Matteo Boiardo's epic Orlando innamorato. Princess Angelica of Cathay (China) is distressed by the Muslim Tartars at the city of Albracca. Riding to her rescue are the French, the Indians, and several other Muslim armies including King Sacripante of Circassia. She thinks all this isn't good enough and escapes to find the missing Christian champion Orlando before returning to be rescued.
Twilight: Bella Swan is ineffectual against a group of rapists and Edward must swoop in to save her. Prior to this, Edward had to save her from a careening truck. Later in the book, she is ineffectual against a vampire, and Edward and his family must swoop in to save her. Subsequent books have the same formula, right down to warring factions — werewolves and vampires — putting aside their differences to save Bella. Bella herself is absolutely useless in a fight until she herself gets cool powers.
Like Sookie Stackhouse Bella is actually the only human with enough bad luck to attract both werewolves and vampires (and various deadly situations) that are impossible to kill or harm unless by other supernatural creatures. One of the reasons of her insistence to become a vampire (aside from spending eternity with her beloved Edward) is to avert this trope. Like she says in the first book: "I can't always be Lois Lane. I want to be Superman, too."
In the movie at least, Bella attempts to fight back against the rapists and maces the vampire before running for it. While neither is winning a battle, it's at least some form of self-preservation.
Let's just say that it's realistic insofar as, a lot of the time, Bella could not realistically be expected to fight off vampires and so on. Everyone else's willingness to sacrifice themselves for her, on the other hand . . .
Though reasonably competent, actor Lee Nicholas (in Tanya Huff's Smoke And Shadows series) seems to have an attraction for evil forces that want to possess his body, hold him hostage, and otherwise put him in peril—perhaps because the series protagonist has a crush on him. At one point, Lee actually says that he's "getting tired of being the designated damsel in distress".
Buttercup in The Princess Bride spends almost the entire story waiting for her true love to come save her. She's in this mess because she gave herself up to save him — and he did promise he'd always come for her. Of course, The Princess Bride is an Affectionate Parody of swashbuckling adventure stories.
In House of Leaves, Pelafina writes in her letters that she is this character, and that her son has to save her from being locked up in the mental institution.
Esmeralda in Notre-Dame de Paris. Her mere presence is the catalyst for all the action in the book. Victor Hugo kind of rips into this trope by having Esmeralda pine for her knight in shining armor, only to be hanged by him in the end. Had Esmeralda been a little more proactive about her own fate, maybe things would have worked out better for her.
In The Phantom Tollbooth, Milo's quest rapidly turns into one to rescue the princesses Rhyme and Reason from the Castle in the Air. Once Milo reached them, there was a huge group of very PO'd monsters racing towards them, so running was the only option any of them had.
In The Moomins, Snork Maiden, and being so pathetic has made her the least popular character.
She often does it on purpose, since she fancies herself as a romantic heroine. She can be quite undistressed when she wants to.
Elayne, Egwene and Nynaeve from the earlier books of The Wheel of Time. They have a strange ability to get shielded, tied up and locked away only to be rescued by someone, though they did manage to get themselves away from the Seanchan in Book 2. Plus the time they actually berated Mat for saving them. They do get called on that later on by Birgitte however, who tore each of them a verbal new one and forced them to apologize. They'd also broken themselves out of there when Mat showed up.
Although Terry Pratchett insists he's unable to write characters like this, Ginger in Moving Pictures spends her short-lived Holy Wood film career playing the role of one Distressed Damsel after another.
He's clearly forgotten Violet Botell in Hogfather. Susan does lampshade it by berating her in her mind for her intentionally helpless behaviour.
The Silmarillion: Played straight with Finduilas, killed by the orcs, Niënor Níniel (when Glaurung wipes her memories off). But very much subverted with Lúthien: when imprisoned by her father, she frees herself. Although she is then captured a second time and needs some help to escape, she then proceeds to almost single-handedly free her lover Beren (and a number of other prisoners) from Sauron — yes, that Sauron. Another example from Tolkien is Celebrían, the wife of Elrond, killed and possibly raped by the orcs.
Emma von der Tann in The Mad King meets Barney Custer when he sees that her horse ran away with her.
Both Sanoma Tora and Tavia in A Fighting Man of Mars. Sanoma loses her spirit entirely, which is evidence enough that she is not, after all the Love Interest.
Judge Dee's cases often include at least one of these young ladies; ranging from vagabond thieves, to reluctant prostitutes to innocent young ladies of gentle birth. However they are seldom quite helpless or useless.
Wilkie Collins' Victorian novel The Woman in White (1860) features the character Laura Glyde (nee Fairlie), who is the embodiment of this trope. She's got the emotional strength of a Kleenex.
The interesting part is that Marian Halcombe, her half sister, is an amazingly strong character for a Victorian novel, almost an Extraordinarily Empowered Girl by the standards of the time. While Laura is the epitome of blushing Victorian beauty and fragility, Marian is described as "ugly", even having a slight mustache on her upper lip. Maybe this is a case of an Ugly Tomboy and Girly Girl.
In Wen Spencer's Endless Blue, Paige is captured by Mary's Landing and Turk must come to her rescue. Also Eraphie did not flee of her own will but was captured by Hardin; Mikhail comes to her rescue as soon as that becomes clear.
Diana Mayo, heroine of The Sheik. She's kidnapped by a rival Sheik, forcing the titular character to rescue her, during which he realizes he's fallen in love with her.
In the Dragonlance series, Laurana becomes this after being captured by her Arch-Enemy Kitiara and having her love interest Tanis Half-Elven try to rescue her. Partially subverted though in that Laurana no longer trusts Tanis as he has been Dating Catwoman, refuses his help and ends up breaking free on her own. Though she does end up needing Tanis's help to complete her escape.
In her "Majyk" trilogy, we first have Mysti in Majyk by Accident whose only source of distress is her Welfin relatives and who bullies Kendar into marrying her so she can leave the "jolly greensward ho" and stop skipping around like an idiot and her only REAL distress is when the curse hits her after Kendar refuses to follow through with a promise he made during the wedding vows.
In the second book, Majyk by Hook or Crook, we have not only Mysti who has become the swashbuckler with a secret identity, A Blade for Justice (and prefers to be referred to by his/her full name), but we also have Anisella, who wears nothing but chain mail, has a black belt in helo kiti and a green barette in po kipsi, and crumples like a McDonalds napkin when even barely brushed by wool... or any other fabric.
The third book in the trilogy, Majyk by Design, gives us a male example in Prince Boffin who has been turned into a toad but also gives us great parody in Kendar's aunts (mercenary swordswomen)and his soon-to-be sister-in-law Dulcetta who, although she is generally the TYPE of girl who would fall into this category, actually kidnapped the man whom everyone thought kidnapped her and hatched a scheme with him to write romance novels. When the main characters find her she is heard screaming for help with the help of a metric ton of Purple Prose and while she is recounting to them the story of what happened runs off to write when the characters paraphrase her cries as "Help me". She thought it was perfect. It also comes to pass that her mother, who raised her to be a docile, dependent woman, was a barbarian swordswoman herself and only gave it up because she preferred regular bathing.
Jez is kidnapped at the beginning of the second Kingdom Keepers book, setting the plot in motion.
Inverted in Journey to the West where Sanzang, the only human of the group, and a man to boot, is often kidnapped by the newly introduced Big Bad of each chapter.
In Andy Hoare's White Scars novel Hunt for Voldorius, the Bloodtide tells the White Scars and Raven Guard that Malya is being subjected to being made a new Bloodtide, and begs them to rescue her.
Ginny Weasley in Harry Potter and the Chamber of Secrets, although no one realizes it until near the end. And she did attempt to save herself by throwing the diary away first, stealing it back only because she was afraid of being outed.
In the first book of the Time Scout series, Margo ends up in a 16th century Portuguese prison. In the third, Birgitta is saved by Skeeter from a beating. In the fourth, Birgitta is saved from gang rape and murder. In the third and fourth, Ianira is in the hands of Jack the Ripper.
Lampshaded in Soon I Will Be Invincible, where it is noted the Corefire has the requisite "reporter girlfriend who always needed rescuing."
Subverted most of the time by Jenna Heap in Septimus Heap, as she usually manages to get safe by herself.
In L. M. Montgomery's The Blue Castle, Valancy foolishly goes to a dance where drunken men start to harrass her. Barney Snaith arrives in time. The main character in Anne of Green Gables is saved by her future husband from a catastrophe resulting from her attempt at impersonating Elaine the Lily of Astolat from Tennyson's poem. Hilarity Ensues.
In Teresa Frohock's Miserere: An Autumn Tale, Lindsey is in Hell. Lucian realizes he must open a Gate, which has been forbidden to him, to rescue her.
Amy Goodenough in the Young Bond novel Blood Fever.
Agnes and Antonia both get their chance to fill this roll in The Monk. One will live to be rescued, one will not.
In Poul Anderson's "A World Called Maanerek", Sonna is captured with Torrek. While Torrek is turned back to Wanen by removing his new memories and restoring his old ones, the ship decides to use Sonna as a "tension release" by lobomotizing her and letting the men rape her. Wanen, his memories not so gone as they thought, rescues her before his own escape.
Also by Wen Spencer, the cover to A Brother's Price features a man carrying a limp woman. Said cover is misleading; that scene does happen, when Odelia passes out in a stream after being beaten by attackers and Jerin fishes her out, but the women of that universe, Odelia included, are anything but neutral, and that is really the only case in the book where a woman needs to be rescued.