Boerum Hill is a great neighborhood, and is basically contiguous with Cobble Hill and Carroll Gardens. Many people around here refer to the three together as BoCoCa. Compared to Cobble Hill and Carroll Gardens, most of Boerum Hill is farther from the major retail/eating corridors and subways, though if you're on the east side you're good for subways, and if you're on the west side you're good for both the subways and the commerce. The area between Smith and Court used to still be considered Boerum Hill, but most people think of it as Cobble Hill now-- I think because Boerum Hill is just thought to be 100% residential.
The only issue one might have with the west side of Boerum Hill is the Gowanus Houses, which is the only housing project in BoCoCa. There are definitely some undesirables hanging around, and this is the least lovely part of the neighborhood (the towers are particularly ghastly). But this wouldn't be a significant consideration for me-- crime rates are still very low.
Another plus of the Cobble Hill and Carroll Gardens parts of this neighborhood are the community parks. We didn't like Boerum park nearly as much.
I think if you're looking for family friendly neighborhoods, BoCoCa is great. The only issue is that rents are going up. Two bedroom places are more in the $3500-4500 range, though you could do a one bedroom with a den for $2,400-$2,600 (usually the den is railroaded with no window). A somewhat cheaper alternative that is equally as family friendly is Park Slope
