This is based on opinion. Please don't list it on a work's trope example list.
Nightmare Fuel / The Baby-Sitters Club
Claudia sits for a family in which the husband abuses the kids. After Claudia and her mother rescue the children and their mother, Claudia gets a phone call late at night. It's the abusive father, wailing at her to give him back his children.
In one of the early books, a serial thief was breaking into houses in nearby towns, making calls to prospective targets and instantly hanging up if anyone answered. The club starts getting calls where the caller hangs up once they answer, and later in the book, Claudia's babysitting and realizes that there's a prowler outside. It's one of the guys from her class, being a moron, and the calls were from another guy who couldn't work up the courage to ask her out, but it's a pretty Adult Fear-ridden situation.
Then Dawn's brother Jeff (both are stranded) theorizes they could be in Nova Scotia for all they know and another charge refers to the book Baby Island.
And Dawn and Jeff's mother being terrified that even if the kids are found alive, her ex will sue for custody because of the whole incident. When you consider how many perfectly decent and good parents have lost custody for less, it's a very legitimate fear.
Stacy and the Haunted Masquerade is far scarier than most of the mysteries in that the villain is a legitimate escapee from a mental institution who is revenge-crazed.
One of the sitters from the rival Babysitters Agency permitted Jamie, age 4, to go outside in the winter without gloves, hat or scarf and didn't bother to supervise him, resulting in him playing in the street. The BSC manage to get him kitted out and safely into his fenced-in backyard, but all of them are horrified at what could have happened if they hadn't come by.
Mary Anne Saves The Day features a little girl coming down with a high fever, imagine what could've happened if Mary Anne and Dawn weren't clear-headed and capable enough to take care of her.
How about the bit in Dawn and the Impossible Three where Dawn wasn't informed that a charge is allergic to chocolate? If Mallory hadn't swooped in and took the treat, there could've been a dead (or at least extremely sick) two-year-old...
And more! Everyone fears that Buddy Barrett has been kidnapped, only for it to turn out it was his father taking him out for the day because his ex-wife is a moron about the visitation schedule. Scary since real-life divorced parents might let their ill feelings towards their spouse take over...
In Kristy and the Copycat, Kristy joins the school softball team, and is bullied into taking part in a hazing ritual for the new team members to spray-paint graffiti over an old shed. Subsequently, they find out that a fire started that night and a man was injured; and assume it must have been somehow their fault, since spray paint is flammable and they'd left their paint cans behind. It turns out the fire was actually caused by other students, who deliberately started the fire in order to make themselves look like heroes for putting it out, but it blazed out of control. Either way, it's pretty scary to think about how a seemingly harmless prank can go dangerously wrong.
One book that dealt with Jessi discovering that a fellow student at her ballet school has an eating disorder was very unsettling, given how common eating disorders are among young dancers in real life - even girls of Jessi's age (11-12.)
Mary Anne + 2 Many Babies features an Egg Sitting plot, with Mary Anne coming to class one day to find two of her classmates in tears because they lost their egg in a park. Mary Anne is unsettled when she realizes how easy it would be for a real child to go missing, and it's an effective source of Adult Fear for the reader too.
The Mystery Special Babysitters Beware! involved the girls sitting for a family who have just moved into a magnificent ancestral home in Maine. They discover that the family's "servant" is actually a distant relation scheming to force them out of the house so he and his wife can have it for themselves. At one point, the girls find he wrote down his entire evil plan, and are horrified to discover that said plan includes starting a fire in one of the children's bedrooms.
Happy Holidays, Jessi is rife with this. The main arc is that before Christmas and Kwanzaa, Jessi, Becca, and Squirt had gone shopping with Aunt Cecelia and gotten involved in a car accident on the way home. Squirt ends up in the hospital for the majority of the book, and Jessi thinks it's somehow her fault because she took off Squirt's car seat belt right before the accident. Granted, Aunt Cecilia told her to; the situation involved everybody blaming each other for awhile.