Well, Boston is cheaper (by a bit) than New York, and possibly a bit more expensive than Chicago - but that depends on lots of personal factors.

If you're talking about living smack in the city, all three areas are horrifically expensive - though NYC is far and away the most expensive for housing of any type.

I've lived in the suburbs of Boston and Chicago (and in the far suburbs of NYC - a LONG time ago), and there are plenty of areas that are much more reasonable than the city center. New York has the best mass transit (assuming you have to commute into the city to work), with Chicago next and Boston clearly third.

What brings you to the US? If it's a job, you may have to work around the location where you'd be working (i.e. city center in one city vs. suburban "campus" in another).
Cheers,
Bev