We've been there for NYE the past couple of years...The resort has parties for the kids and teens. This past year, there were three different parties -- one for younger kids (don't remember the exact age cut off

) one for pre-teens, and one for teenagers. There were no planned adult activities.
Many of the HH restaurants have special events for NYE. We just cook a "nice" dinner in our villa (using Chef Gordon's crabcake recipe for "atmosphere") and just go out on our porch at midnight to welcome in the New Year (as do many other members and guests).
On New Year's Day, there has been a touch football game on the beach, like on Thanksgiving.
It's a pretty laid-back celebration, but we love it!
As far as the rest of the island, some shops and restaurants are closed, but many wait till after January 1 to take their winter breaks.