If I saw a little dead kid it would freak me out no matter how cute he was.
It's an interesting conundrum, isn't it? The Ghostly Trio and Spooky look far scarier in most of their appearances, but have to work to frighten people; yet Casper, who looks totally non-threatening, can freak the norms just by smiling and saying "hi." There's clearly something else going on that we the audience can't sense.
It's kind of fun to think about. Maybe it's because he looks the least human or something. Maybe it's his featureless face. Or maybe he just carries a really freaky aura about him. Maybe it's even the fact that he doesn't want to be scary, that he tries to go against his natural desire, and that makes him seem so off that people are more freaked out.
One cartoon had a humorous inversion of this: The Ghostly Trio are frustrated in that they can't scare anyone anymore, as the local animals have grown used to ghosts thanks to Casper. They try to rectify this by forcing Casper to be mean, with disastrous results.
Even more interesting, babies and small children, who are normally quite easy to scare, are very rarely shown to be afraid of Casper, only adults and (usually older) teens. And even adults can overcome their fear fairly easily, given cause.
Unlike the movie adaptation, Casper doesn't really seem to have a backstory concerning when he was alive. It just seems like he has been a ghost since "birth".
Originally, he was a dead child; however, the very idea of this made people Squick, so when Harvey acquired the rights to the character, they changed it; the ghosts were retconned into being magical creatures rather than the spirits of dead people. Harvey even went on record saying that Casper is a ghost simply because he was born to a husband-and-wife ghost couple. ...Wait, WHAT?!
In other words, a phantom pregnancy?
Casper's "family"... just how many relatives does he have? He has the Ghostly Trio as uncles and Hairy Scary reoccurs in various animated series, but is he related to any of them?
What happens to the corpses of the dead? Dr Harvey will have a lot of explaining to do if his body is ever found in that trench...
If ghosts are dead people with unfinished business, then why isn't Casper's father a ghost? He never got around to bringing Casper back to life.
One must assume his business was the creation of the Lazarus Machine, which satisfied him enough to pass on.
I bet Casper's father is more of a douche than goody-two-shoes Casper gives him credit for.
I just thought he was alive and in an asylum.
Regarding dead people, Carrigan dropped off a cliff, which removes difficulties with corpses. As for Dr Harvey, maybe the Ghostly Trio dealt with the corpse before they headed back. Apparently the Lazarus machine can give you a new body, it seems.