I'm sure it's vague on purpose, and that probably I'm not supposed to think there's an actual bad thing he did, just imagine that it's worse than you can imagine. It's just that my "worst" has been reset to The Great God Pan and Dorian wouldn't *do* that!

The reason I don't think it can just be generally infamous sexual conduct is that Lord Henry also does that and Basil seems to think Dorian is way over even Lord Henry's lines. Although Henry was married, which means he can't be misleading anyone into thinking he's a marriage prospect, so that may be a big part of it.