I believe it was mstevens's wife that was mugged on her scooter, but she was well off the tourist track and probably mistaken for a local. Apple was mugged pretty badly, but she's definitely a local. Probably the best bet is to look as touristy as you can when wandering in unfamiliar neighborhoods. You'll get reamed when trying to bargain for any purchases, but at least no one will touch you. A big foam hat might work.
No, there's no high crime in Cozumel and probably not even any dangerous neighborhoods. I don't say that from experience, I say that from knowledge of how small the island is and how tourist-dependent they are. That said, it's a Latin-American country which means there's always a "macho" attitude present among the least worldly of the male population and any female walking along by herself late at night in the wrong part of town could quite possibly be asking for trouble. If you have a large-sized male escort/boyfriend/husband who is good at avoiding conflict, no problem. Otherwise, well, I defer to what others have to say about inland Cozumel at night since I don't have any personal experience wandering around as a lone female.
