You could also consider communities to the north, starting in Northwestern Westchester: Yorktown or Croton, but you would not likely want Peekskill from a school perspective. Commuting on the Hudson Line train from Croton-Harmon (15-20 minutes from just about all of Yorktown) on an express will get you to GCT in about 45 minutes. Then, add another 20 or so to get downtown via subway.
If you could tolerate a slightly longer commute via the Harlem Line trains, you could look to Mahopac or Carmel in Lower Putnam County. There would be more in your price range there, and it's a typical suburban environment, with good schools. You could drive about 15 minutes to the train at Goldens Bridge, though you would need to secure parking, which is not always easy in Westchester.
Since you would be in Lower Manhattan, you could also commute from Monmouth County, NJ via the SeaStreak. Red Bank would put you about 15 to 20 minutes from either dock, and there are express boats that make the trip in about 45 minutes or so, just about an hour to get to Lower Manhattan. There is shuttle service to the WFC from certain boats at rush hour, so that would save transferring to the subway or bus from the pier. It's significantly more expensive than the train, however, about $650/month, assuming 20 round-trips.
All the world's a stage, and all the men and women merely players: they have their exits and their entrances; and one man in his time plays many parts, his acts being seven ages.
~William Shakespeare (As You Like It Act II, Scene VII)
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