I am currently finishing up my undergrad at U of R, and I live off campus. I can tell you that while it is more convenient to live on campus, you can definitely get by without a car by living along one of the bus lines (particularly the red or gold lines). The red line runs from U of R main (river) campus to Eastman school of music, and runs through South wedge, park ave, and other neighborhoods about once an hour from 6 am-2am. The gold line runs thru the 19th ward (not a great area) about every 30 minutes. I live in southwedge on the red line, and although I have a car I almost never use it to get to school.
I currently live in an appartment in south wedge about a 30 minute walk from campus, 2 blocks from a red line bus stop. My rent is $845 a month split 3 ways, so its really cheap. Its in a pretty safe part of town, a short walk away from great bars and quirky hippy shops. We got our apartment through Southwedge properties LLC., they have lots of them in the area.
Cost of living in Rochester is really cheap, which is good! the best off campus places to live are probably along the red line or behind strong memorial hospital. If you want something a little nicer, there is an appartment complex called Rustic Villages near campus that has buses running to it a few times a day. Its a nice facility, with a gym, laundry, security, and pretty new buildings.