Cozumel - January average low 68, average high 82. If you go to weather.com and click vacation planner, you can get average highs and lows for most popular vacation spots in the Caribbean. I'd also check the average temps for the port of departure. May be a little cool your first and last days
We cruised that area this January, leaving out of Houston, TX. It was cool in Houston, but by the time we awoke on the first day it was warm. It was nice until the last day at sea. We were cruising into a cold front. Otherwise, the weather was amazing.
We went Jan 25 to the East. Carb. weather in upper 70's to 80's great. Warm and very swimmable weather on ship and on the islands. Ocean water might have been a little cool, but not much. You can get very burned so be careful I used 15 sun protection and got a nice tan. Shorts and capri weather.
Leaving from the U.S. it will be somewhat chilly, o.k. chilly for us Floridians, but the farther south you go the warmer it will be. As another poster said, it will be in the low 80's in most of the ports.