It is a wonderful, and very wise idea to vacation in the Caribbean in January. It's the off-season, so you will get great discounts and special packages.
January is a great time for a Caribbean vacation. The Caribbean is close enough to the equator that it does not get too cold. Also, it is the off season for tourists, so prices will be much cheaper and the crowds will be a lot less.
Yes, it is. Because of lower demand, prices are lower. But the Caribbean is warm year round, so swimming and snorkeling are still options. There are also fewer crowds to deal with.