From NY I'd look at Turks/Caicos - esp. Provo (Providenciales) Grace Bay is one of the nicest beaches in the world. Dive operators will pick you up, it's all boat diving so you should be able to find a guide/DM buddy. Best diving is 45mins. out at West Caicos or French Cay but there's also decent diving at NW Point nearby. There's three AI resorts, 2 of them have kids programs. The third, Verandas has villas for rent. There's also a lot of condos for rent on Grace Bay Beach. Provo is very nice, very picturesque and very
pricey. IDK if there'd be enough there to keep the kids occupied unless they like beach activities, boat rides etc.
Beautiful by Nature - Turks and Caicos Islands - Turks and Caicos Tourist Board
Or Grand Cayman - Seven Mile Beach is also extremely nice. There's quite a few non-dive things to do there as well - butterfly farm, gardens, etc. There's a lot of condos for rent along Seven Mile Beach - most are waterfront/oceanview. For the kids, there's recognizable food, Burger King, Subway, Dominos etc. on the island. It's equally pricey - maybe even more than T/C - the CI$ is fixed at .80 to the US$. When the cruise ships are in Georgetown (almost daily) it's a good time to be elsewhere. You can see them moored out in the harbor from south SMB.
http://www.caymanislands.ky/default.aspx?homeredirect=true
Along SMB dive operators pick you up for free. Most are 6-8 divers max. per trip so you may not need your own DM. Boat dives are 2/$100 with anyone. One new dive there is the Kittiwake wreck, it's purposely sunk in 70' - you can penetrate the 1st three levels - it will be with a DM - mandatory there. One thing to note, first dives there are often deeper (100') down the wall.
Sometimes there's a shallower option also.
The Official Website for the Kittiwake Shipwreck - Home - besides describing the wreck dive, there's a pretty complete list of Cayman dive operators.
A third option might be Curacao. They have a lot of nice family beaches - most rent watertoys etc. Downtown has the famous shopping district and an open air mall. There's some historical monuments - an old fort (now a shopping center/restaurant) etc. There's the Ostrich Farm and Seaquarium for the kids - think small SeaWorld. There's also one of those fake submarine rides if they want to see what Daddy does.
Curacao Travel- Caribbean, Curacao, Dutch Caribbean
The Dive Bus will pick you up for escorted shore diving at any resort in town, do two dives and back by noon. Most of the resorts have an onsite dive operator so if you rent a condo nearby it's a short drive to one for boat diving. Most of the better beaches also have on-site dive operators and shore dives. At many you can rent a guide as needed. My friends wife used to be one there.
These 3 locations are shorter flights from NYC. With 2 kids under 6, that may be a consideration.
From Newark there's a United non-stop that's 3h45m to Provo. Return is 3h55m. LaGuardia (heard it's re-opened) is 6H out, 7H back - connect in MIA. JFK is about the same. All flights are in the $400-475 range.
American has 5-1/2-6hr flights to Grand Cayman from JFK or Laguardia. All connect thru MIA. All $450-500.
Curacao is 8-9 hrs. each way thru MIA. From any of the airports. But in the $650-700 range. You might make up some of the difference since everything else is cheaper.
I get all this flight info here:
Matrix - ITA Software - it's the database all the airlines use. You can't book with them, do that direct.
You're kind of doing this last minute so I'd book with dives also when you know what your plans are. At least on Provo or Curacao - Cayman there's a lot of options.