I only know one person in Boston who lives on their own, and they make ~70k/year, but if your trust fund has $1500/month, go nuts, I would, except I don't have a trust fund.
BU is kind of a pain because it's not near any major subway routes, it is, however, on the 1 bus line, which runs about every 5 minutes from 5am to 1 am and goes straight up Mass Ave to Harvard Square in Cambridge in about 25-30 minutes, depending on time of day; I used to use this route all the time, and found it to be extremely reliable. Pretty much the only reliable bus route in the city. I'd see about living in Cambridge. I live in Inman Sq., which is about a 10 minute walk from Harvard and Central Squares, where you can pick up either the 1 or the red line. Best part: I have two awesome roommates and my rent is $400/month for a decent apartment with a good sized room.
For Tufts, I'd look to Jamaica Plain, Medford, or Somerville; basically somewhere along the orange line. One thing to remember though: much of the orange line was installed along a route that was originally planned as a superhighway that was to cut straight through Boston, so naturally, the neighborhoods it runs through tend to be slightly unsavory. That said, I spent three years in the sketchiest part of Jamaica Plain, and next to Inman, it is my favorite neighborhood in Boston.
Harvard. You're gonna pay. It's really only along the green line, which happens to run through the nicer areas of Boston and points west. You'll probably end up living in the Longwood area or around Mission Hill. I lived on Mission Hill for a few years, and it was pretty much a student ghetto when I lived there, but when I went there recently, it's changed a TON, probably for the better.
In case you can't tell, I don't mind living in sketchy and dirty neighborhoods to save a few bucks. I've lived here for going on 9 years now, so feel free to ask any questions about specific areas, chances are I can tell you something about them.