Wait, are we talking STDs, or STIs? I'm assuming we're just referring to both as STDs...
It depends really, I probably wouldn't make a big deal about it. I'd get checked out, have her checked out at the clinic, if she had something cureable, then alright, take some medication and don't do any of those under-the-cover things, problem solved. Besides, I'd use protection so my chances would be slighly lower if nothing else.
If they had an STD though, like, legit STD and not an infection, then I'd have to think about it...
If we were dating, they had recently become my significant other, etc. Than I would probably tell them "hey, I like you and all, but this isn't a risk I'm willing to take." because I wouldn't want to risk a LIFE-LONG DISEASE on someone who it may or may not work out with.
On the other hand, I'd be willing to stay with them if we were, for instance in a very serious relation status, or married. I could probably deal with it, as the majority are preventable if treated carefully and such.
HIV/AIDs though... I'm really not sure. I would stay with them, but I wouldn't even want to risk infection. (Yeah, I know the only real ways to get infected is blood transer and "naughty acts" since you have to get a large amount of the virus to infect yourself.) I'd make it work out, but I wouldn't have sex, and I'd be super-cautious about cleaning a wound. It wouldn't matter how much I loved them, I just wouldn't be able to risk sacrificing my own life for something as trivial as sex, or some kind of symbol to show I would go there and risk my own life like that. (again, don't get me wrong, I'd jump into burning buildings to save people, but diseases that slowly kill me off? no thanks.)