Could you sugest being a study buddy? It'd be enough not to scare the blinkers off you by asking her for a coffee but casual enough so you can get to know each other and build up some sort of rapour.
Failing that, how about when you leave the class room, time it so you're walking out together and make a flippant comment about the lesson 'that was a good one', 'could that have been more boring' etc... Something that breaks the ice and eases you in to a conversation and gives you a reason to talk to each other in future. Give that a couple of days to see how it goes and then suggest a coffee/whatever.
Ultimately the worst thing she can do is say no, most people don't tend to completely deflate egos and leave you in need of a counsellor for the next fifteen years in the 'rejecting a date' process.... Just remember, if she does say no, it's not because you're unattractive/uninteresting/not worth dating, you just might not float her boat, but you will float someone elses. It will sting but that's it. (And besides, this is worst case scenario territory!)