There is a lot that one can see and do in NYC, so I would not suggest skimping on the time if you're interested in that sort of tourism and cultural enrichment and whatnot. I suspect you would have no trouble keeping busy for a week or more if you wanted.
As others have said, staying at Normandy Farms to visit NYC makes no sense. It's somewhat like camping somewhere between Seattle and Vancouver to visit Portland. You will spend more time getting to and from the city than doing things in the city, repeatedly. It looks to me like the quickest way in would be via the Acela train from Providence, RI, about three and a half to four hours travel time. Liberty Harbor is the usual campground suggestion. I've stayed at Floyd Bennet Field (in southeastern Brooklyn) not long ago and it can also be entirely workable, though I suspect for most people Liberty Harbor might be a more practical choice overall. Getting from Floyd Bennet Field to Times Squar takes about an hour and a half, perhaps as little as an hour if the bus and subway schedules cooperate just so. Liberty Harbor seems to be about 45 minutes to an hour travel to Times Square.