Only problem will be air conditioning ashore. They like to crank it up so a sweater or jacket will feel good. Only problem on the ship after sundown is if there if the wind is blowing. The ship will be making about ten to fifteen knots which equals nearly 20 mph. If it is heading into five to ten knots of natural wind, that can be a healthy breeze. The weather decks on newer ships are sheltered so you won't notice it. Throw in a sweater, you probably won't need it - however.
Incidentally, the locals observe summer and winter, Labor and Memorial Days marking the shift. The humidity drops during the winter making it feel chilly to those who live there. I remember greeting houseguests (you get a lot when you live in the Caribbean) who would be sweating through their sportcoats before we got to the baggage area while I needed mine to be comfortable.
We cruised in mid-Jan in 2000. No problem in Caribbean, very comfortable on decks even in the evenings, but I am one of those warm-blooded women who do not get cold quickly. New Orleans was one of our ports, however, and they had a cold front run through the day we stopped there. We had packed no warm clothes whatsoever, so we picked up sweatshirts at a local store while in port to get through the day!
I myself am a warm blooded person, it usally takes a bit for me to feel cold.
On the other hand, my wife likes to sleep with a quilt when its 80.
Im sure my wife would like the idea of buying a sweatshirt if it gets cold.
I would bet she will buy one reguardless of the weather conditions.
We sailed Century in February of 2000 to the Western Carribean (Jamaica, Grand Caymen, Cozumel and Key West). The weather was comfortable bordering on hot. The first day was at sea and that was almost chilly with overcast sky and a stiff breeze. After that it was pretty good. We did 7 Mile Beach on Grand Caymen, and the water was cool. The hottest day was the second day at sea going from Cozumel to Key West. We were off the coast of Cuba, and it was hot! But compared to the snow and cold we left in Connecticut, it was nice. No rain at all.