I went to BU undergrad, and I gotta tell you, most of the apartment/dorms on campus (main campus on Comm Ave) are a bunch of crap. The slight exception to this rule are the brownstones on Bay State ROad.
Here's a list of decent places:
1) Brownstones on Bay State Road (student apartments, but beautiful, quiet location)
2) Maybe, possibly, the student apartments in South campus. But maybe not. Lots of loud undergrads.
3) THe SOuth end (where the hospitals are located) is a beautiful part of town. It's expensive though.
4) The Back Bay, Beacon Hill, and Brookline are wonderful parts of the city, but you'll pay an arm and a leg.
Basically, everything in Boston is expensive. No doubt. But the good thing is that if you live in Boston proper, you'll be able to get around wherever you need to go without too much trouble. Boston is a really small towm (in surface area), so it doesn't take long to get from the North End to the South End (you could walk it). That said, I would really stay away from Allston, Brighton, or Kenmore Square. Those are all student slums and will depress you.
If you want to explore options outside the dorm, go to <a href="http://www.boston.com" target="_blank">www.boston.com</a> (I think that's the address) and find The Boston Globe. They'll have a listing of apartments and parts of the city, to give you an idea. Also might want to check out a map of Boston to see what's around.
Good Luck.
PM me with more questions.