Hi!
We just returned from HHI and I asked that question to a CM. Apparently there is much to do on HHI in the off-season. Both pools are open and are heated to 84 degrees. I think she said the shuttle to the beach house still runs, but the snackbar at the beach house is closed. They concentrate on more activities that are indoors. More arts & craft type of things, bingo, etc. are emphasized. Bicycles are always a good idea, especially with the extensive bike paths on the island.
The dolphin cruises do run, according to Scott, our guide on the cruise we went on. They are booked on a "as needed basis". Golf is available. Restaurants are open. Just about all activities are available except going in the ocean, unless you don't mind cold ocean temperatures. They are trying very hard to make HHI a year-round destination.
I hope this was helpful.