How long are you staying? On Sunday, the only one I know for sure is running the double deckers is the Oyster Bay line that you can take from Jamaica. During the Weekdays, there are a few more that you can pick. BTW if you're near the NY area and you like double deckers, NJ Transit also has double deckers, and they're even less predictable.

Oyster Bay has the added benefit that there is a small railroad museum there and also has a beach, though the beach is not that great.

Also, a bit further out in Long Island, is the Long Island Steamers which your kids will surely enjoy, and tomorrow happens to be a run day. http://longislandlivesteamers.org/index.php

Also, regarding LIRR Double Deckers, any train going to Montauk (or any other unelectrified line) are double deckers.