I would definitely second the suggestions of Logan Square and Lincoln Square - both affordable and easy around from using the Blue or Brown lines. I haven't lived in Pilsen for 10 years or so, but I think it is a great area as well.
To look at crime rates in specific neighborhoods you can look at a couple sites:
http://12.17.79.6/
http://www.chicagocrime.org/
They both have maps so you can see reported crimes around any apartment you're looking at. Safety is best judged on a block by block basis in any part of the city.