PayPal is generally a pretty safe way to send money, because not only is there a money trail, but PayPal has pretty good fraud and buyer protection. Furthermore, if you fund your PayPal account with a credit card, then you get the additional protection provided by your credit card bank too.
If you use PayPal, there isn't really a way for her to take more money from you (unless she knows your PayPal username and password); the worst case scenario is she can take the money you give her and run off with it (and not rent you an apartment).
Perhaps it would be fair to offer that you'll pay a portion of the rent in advance, and the rest once you arrive in person and can verify the apartment is as advertised.
Also, PayPal has the concept of a Verified Account. This doesn't necessarily mean you shouldn't trust someone who isn't verified, but if they are verified, that may help you feel more comfortable with the arrangement.