I believe all official sources (the scholastic website, the book covers) say it's "Baby-Sitter's Club" (and I think it's the first book where Janine comments that really, it should be "Baby-Sitters' Club"). Wikipedia uses "Baby-Sitters Club." We should really decide on one variant because, in article names, Baby-Sitter's Club and Baby-Sitters Club will go to entirely different places. Besides, it'll look neater.
I guess I would lean toward not using the apostrophe, on the basis that even the books themselves admit that it doesn't make that much sense punctuation-wise, and I think that the majority of people probably don't use the apostrophe when writing the series name. The problem (but also the bright side) is that most of us just use BSC all the time so the apostrophe is irrelevant. But on the other hand, the apostrophe IS in the official spelling.
Thoughts? Could we get away with doing a poll on the babysittersclub (again, no help!) LJ community to ask what the most common usage is? Although I'm not even sure if I could say how I most commonly write it.