I'd say one of the boroughs - Brooklyn, or even better, Queens - rather than Manhattan, which is what most people are talking about when they say New York City. (That's assuming when you say "New York" you mean the city and not the state.) So I'd go for Astoria or Jamaica (in Queens); Queens is much larger than Manhattan, which allows for actual houses (as well as apartment buildings) and a lot of residents have cars, which would be a necessity for a family that size. My dad lived in Jamaica with three roommates in the early 70s for a few years, so it had to be quite affordable.