As others have said, know your market. In fact, I think that this is far more important than anything else. If you know where the price range should be, it really doesn't matter who starts first. It keeps you from throwing out or accepting an offer that is far more than what was needed while also ensuring that you don't make an offer that is insultingly low.
With that said, my preference is to have her tell me what she needs. In my experience, girls who are in competitive markets will often start at a reasonable number to begin with and the ones in tighter markets (like NYC or L.A.) who quote a ridiculously high number were never going to leave for reasonable money anyway, so this minimizes the amount of time I have to burn with pointless negotiating.
For a girl in a competitive market who throws out a high number, my responses vary among things like:
"Honestly sweetie, that's a little high for this area, even for a girl as attractive as you."
"I'd really like to take you out, but that's so much higher than my next best option that I just can't get myself to pull the trigger. Is there any way we can bridge the gap here?"
In my experience, 19 times out of 20, a girl in a competitive area who initially high balls will counter with a more reasonable number. In a few instances, For the 1 out of 20, you just have to let her know that you understand and then cut her loose. In a few instances, I've had girls leave and come back later with a more reasonable number, no doubt after they exhausted efforts to earn more elsewhere.
When I am forced to make the first offer, I might low ball a little, but not by much. The idea is to get to a place that everyone is ok with, not to try and nickel and dime the girl over $20-50.