I went to BU for undergrad, and am currently living within walking distance of Harvard. The best option for what you're looking for would be Allston. It's one of the less expensive neighborhoods in Boston, it's walking distance from BU, and there's a bus going straight from Allston to Harvard to see your friends.
Of course, every rose has its thorn...Allston has a well-earned reputation of being trashy and filled with drunk, drugged-up college kids. While this is all true, there's a lot to like about Allston! There are lots of great (and diverse) restaurants, every convenience you need is at your fingertips (laundry, supermarket, ATMs, etc), and it's honestly not that trashy. It's not the classiest place to spend a Friday night, but it's perfectly fine for living purposes. BUT A WORD OF CAUTION: DO NOT LIVE IN "LOWER ALLSTON"! Public transportation (and civilization) is almost non-existent there. You want to live as close as you can to one of the following streets: Commonwealth Ave, Brighton Ave, and Harvard Ave.