H.S.
I have gone on the 1st week of January. We were surrounded by Santa's. Apparently they all go on a cruise together for a Santa convention after Christmas time.
Carnival is the least expensive, and extremely kid friendly. We did 5 nights, and our kids tickets were only $99 each. If you don't drink a lot of alcohol, a cruise can be very inexpensive. Once it's paid for, it's paid for.
We went to Grand Cayman and Cozumel and we had a fantastic time. Grand Cayman was the most beautiful place I'd ever been. The best part about the cruise is you don't have to spend tons of money. Eat breakfast on the ship. Pack snacks in your bags to get you through you time on land, and get back on the ship for a late lunch early dinner. Other than a taxi fee, we spend close to nothing while on land. My family of 4 spend $80 the entire vacation. Seriously. Your trip can easily be under $2k!