The transition to any city is difficult, but in my opinion, moving from New York specifically is even harder. I had a tough time adjusting first semester. Boston's a great city, but it's just a lot smaller and there's less to do. You'll find that there's also less diversity in comparison to new york. But you get more for your money in terms of real estate in Boston.
I found my place through Craig's List. Look for apartments in the South End if you want to be within walking distance to school. Just so you konw, the South End is the Chelsea of NY. There's a BU-owned building called Harrison Court that's directly across the street from school, but I hear there's a waiting list to get in there (and rent isn't cheap). I lived one block away from school and was paying 800 per month, not including utilities. BU will only give you enough student loans to cover what they consider to be "reasonable" for a year of living. When I was there, that figure was 1000 per month. This does not include things like internet access and cell phone bills, because they don't consider those to be "necessities". Budget accordingly. I ended up spending an additional 5k from my own savings account in addition to the student loans I got, just to cover basics like utilities, food, the occasional night out, etc. Total including my own money and student loans, I'd say I spent about 15 thousand on day-to-day living expenses (rounding up).
Also, if you call up BU, they have a roommate database, so you can find a roommate from the program if living with a classmate is your cup of tea.