Sag Harbor, maybe Montauk or Shelter Island with the kids
Montauk is probably the most kid friendly town. Ricks Crabby Cowboy on East lake Drive (across from airport) is especially geared that way and has fresh made, fast food type of eats. Lots of room to run around outside. Also great fresh seafood for the adults, and full bar with frozen drinks and whatever. There are quite a few hot dog trucks on the beaches if you want al fresco: all are excellent and have lots more than just dogs. Paddy Wagon in the Gosmans parking lot, Gin Beach Cafe on the beach past Ricks or Ditch Witch on Ditch Plains rd/beach where the surfers go: you can get a picnic lunch and hang on the beach while you eat. Johns Drive in in town, for great burgers and ice cream. You'd be comfortable at any of these places.