For simplicity sake...I would consider looking at apts / home rentals along (or nearby) Haddon Ave in Collingswood, NJ..The PATCO station is just about 1 block away. Sorry, I can't recommend specific places to rent per se, but perhaps a realtor in the area can help...I'm not a fan of Craigslist for this kind of stuff, but it IS an option.
Since I drive, I don't use PATCO to travel into other parts of S. Jersey or Philly...but I did have jury duty last month at Camden Hall of Justice / Courthouse on 5th St.
The City Hall (Market & 5th St) PATCO stop was literally in back of the public parking lot (mostly used for jurors and court employees) where I parked ..which is directly across the street from the Courthouse. I believe the RiverLine Light Rail (which runs from Trenton with several stops along the way before reaching Camden) also runs to that same general area, if I recall correctly.
That area seemed to be safe IMO...I had no concerns leaving my car there or walking in that area. I don't recall seeing any crazies in the area and there was a police presence around the Courthouse area.
I think the commute via PATCO to Rutgers will be fine from what I can see...There will likely be a regular flow of city hall workers and other professionals as well as students day and night...so I really believe all will be well
Best of luck with your studies..and in finding a nice place to live here!
P.S. Check out the map below to give you idea of the proximity from Rutgers Campus to the PATCO Stop (lower right of map):