Following TOS treatment of Greek Gods, witches, and the like, I think it's pretty safe to assume that Santa Claus (along with King Arthur, Hamlet, The Tooth Fairy and others) is one of two things:
A) He's one of Flint's identities. In which case, ah... He's probably dead.
B) He's an alien and always was. Heck maybe that's why Christmas is only one day a year and how he visits all the houses in one night. He's doing the same thing on at least 364 other planets every other day of the year.