Culinary...you're kind of in no man's land since all the close north shore marinas are in "gold coast" country where there are waiting lists for slips and no "livaboard riff-raff" allowed. You might find some marinas in Brooklyn willing to deal with you but that is a long drive to not so great sailing grounds.
I don't think ANY marina on the north shore is going to tell you that they accept liveaboards...though my guess is a few do. If you are determined to live aboard, I would suggest getting a car and driving around to all the north shore harbors and marinas and having face to face discussions with the owners. I would forget Manhasset and Port Washington Harbors...and focus on Glen Cove, Oyster Bay, Northport and (if you're up for a long commute) Port Jefferson. Marinas.com can give you the marinas and phone numbers and directions to the various sites in these towns...or pick up a copy of Dozier's ICW Guide Northern edition.
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