Hope this helps, Jeff Persons ABR http://wesellboston.net/
Your best bet if your on a budget is Allston/Brighton- Kenmore area, though if you have some money to spend you have to live in the Back Bay. Back Bay is very close to BU with more of a professional environment. If your looking to buy I can assist you in that process. Let me know if you have any further questions.
Allston is very popular with younger professionals and college students. A friend of mine lived there for over six years and never had any safety issues, but always check with the police department for a crime report no matter where you decide to live. Cambridge is a great town to live in and very close to BU if you focus in the E. Cambridge area.
As a former resident of Boston I can tell you it is an amazing city no matter what part of it you live in. You may want to consider Cambridge the Harvard Square section is amazing. Full of activity, street sales and performers, etc. If you want to be just outside the city you may want to think about Framingham or Natick. Great towns that are available to the city via public transportation.
Good Luck with your education!
Linda J Sears
Coldwell Banker Ellison Realty