January is past hurricane season, so the seas are generally pretty calm. I've been to the Western in early January - out of Port Canaveral, and it didn't start to warm up until the second day, when we were well south. At sailaway, everyone was wearing jeans and jackets. Once we got to Cozumel, Grand Cayman, and Jamaica, the weather was absolutely beautiful! It was hot, but not too humid or uncomfortable. It was perfect for us! The only downside was fewer daylight hours. It's still early winter, and the sun comes up late and goes down early...like at home. Other than that, we made every port, and it was a great cruise!
We did Sensation out of Tampa on 2/3/03. As Diver said, first evening was a little chilly but we awoke to beautiful tropical air. Lows in Western Caribbean around 68 and highs around 83. Was a great cruise and welcomed break from the Ohio winter.
I guess by the answers it must vary a lot. I've found the western caribbean to be quite rough in the winter when the strong cold fronts come through Texas all the way into the Gulf of Mexico. I've even seen Grand Cayman deny the cruiseships entry and Cozumel cancel all water excursions due to the Gulf stream conditions on several occasions. In my opinion, the eastern or southern caribbean is a better bet, not including Key West.