I would personally have to recommend staying in NY if you want to commute. Commuting from CT is NOT something you want to have to do every morning. Specifically I would say either Staten Island, Queens or Long Island. As far as the 5 boroughs are concerned, Queens and Staten Island is much better than the Bronx or Brooklyn. If I had to rate the boroughs (excluding Manhattan) in terms of all of the things you listed, it would be 1.Queens, 2.Staten Island, 3.Brooklyn, 4. the Bronx. The thing about Staten Island and Queens though, is that the closer you are to the city, the crappier everything is most of the time. The few exceptions are the really expensive gated neighborhoods that tend to be in those places. And plus, the public schools don t tend to be that great, even in decent areas. For my money, Long Island is a pretty safe bet. If you threw a dart at a map of Long Island, 8/10 it would be a very good place. The crime is extremely low, the schooling is top notch and the quality of life is pretty sweet. Plus, the Long Island Railroad (LIRR) takes you right into Penn Station and is a lot cleaner and safer than many other railroads. But Long Island, particularly Nassau County which is closest to NYC, is known for having some of the highest property taxes in the country. However, Long Island is probably the best area in NY in terms of raising a family. Plus, you can probably snag a good deal on a foreclosed home here and make a big profit once the market heats up again. So if you have a safe job, and want a place to call home for at least the next few years, then LI is your bet.
There are some good spots out in Queens. Even relatively far out, there is still convenience to local buses to get to the subway, express buses, and a few branches of the Long Island Rail Road, and you can be in Manhattan from your door in an hour or less using various combinations of those options. As far as schools, again, there are options. You can find some of the better public schools in the city in some of those farther reaches of Queens. Some good options out there if you investigate a bit.
Well New York has relatively low crime rates now thanks to good policing. Connecticut is nice.