Depending on the month and where you are sailing from, temperatures can vary quite a bit. Winter cruises out of Florida will feel very different than winter cruises out of the Northeast.
Most Bahamas islands stay in the 70s even during the coldest months, although occasional cold snaps do happen. On most sailings, especially from Florida ports, kids will still be ready to jump right into the water at the beaches and pools. As an adult, you may be comfortable swimming too, or you can simply enjoy the sunshine and warm sand.
Evenings on deck can cool down into the 60s, so light layers like a sweatshirt or jacket are helpful. One thing that surprises many cruisers is how cold the theaters and indoor venues can feel, so bringing a sweater for shows is always a good idea.
If you are sailing out of New York, plan very differently for embarkation and debarkation days. The first and last days of the cruise can be very cold, sometimes winter coat cold, especially while standing on deck during sail away or waiting in port. Pack warm outerwear for those days, even though you will not need it once you reach the Bahamas.