Since New Haven is its own MSA I dont really consider it part of New York City. Unlike some other satellite cities of NYC (wont name any names) New Haven has its own culture, is a truly unique city and is the heart of the Greater New Haven area that includes roughly 27 towns.
And yes there are thousands who commute to NYC from Connecticut on the MetroNorth Line daily. How many people from New Haven itself commute into NYC I am not sure since it can be about a little less then 2 hours to Grand Central from New Haven. I know lots of people who commute on Metro North from New Haven to Stamford though.
I just rode MetroNorth from New Haven to Grand Central. On a Saturday the train filled all seats in New Haven and became standing room only as it progressed to Bridgeport and then Stamford where it became express to 125th and GC. If that doesn't mean New Haven is part of metro NYC, I don't know what would.