Definitely the UES. I've lived there for three years, and I've never had any safety issues. Granted, I'm a male, but I was even surprised by how nice things were. Even if you go over to the park and run on the southern portions of the main road, early in the morning before it's light out, there's already hundreds of runners out there. I probably wouldn't venture over by myself if I was a girl, but if you have someone to run with you'd be fine.
Three years ago, when the market was a lot tougher for renters, I was able to afford a nice (but small) studio while making only $60k. I'm actually in the process of moving to a real nice 1BR since I make a lot more and want to take advantage of the rents that have come down. Now is a good time to be looking - lots of no fee apartments and many management companies, in addition to lower rents overall, are offering a month free.