My wife and I are seasoned campers be it front or backcountry. We wanted to take our 6 month old daughter camping for the first time in mid September and wanted to give Hither Hills a shot (this place is notorious for being booked solid months in advance). We stayed at a local motel for one night then tried for a walk in spot the next day. We were lucky enough to get one. The campground is nothing special as it is basically wide open with little privacy. We were initially concerned bringing such a young child with the potential for crying (she was a trooper!) but it wasn't a problem with all the families around with young kids.
There is beach access, a playground, and a general store for things you may have forgotten (tent stakes for instance). It is close to town, the Montauk brewery, Montauk Point, Camp Hero, etc. The star gazing was great too being so close to NYC.