Go to: wunderground.com and you can get weather forecasts.
We sailed to the Bahamas in January and while the air temperature was fine (80's) the water was a bit too cold for *me* to be totally comfortable.
CarnivalSensation February 2015
Carnival Dream November 2015
Carnival Fascination April 2014
Carnival Elation March 2011
Carnival Imagination Sept 2007
Carnival Sensation Dec 2006
RCI Sovereign of the Seas Sept 2006
Carnival Miracle Sept 2005
Carnival Glory Sept 2004
Carnival Fantasy Jan 2004
We cruised to Cozumel this past Jan on Inspiration. I have to tell you, the open deck areas aren't much fun in Jan. There is a very stiff breeze and it is cold. One night we watched some little kids holding on to the rails and the wind picked them right up off the deck. Scared me to death.......don't know where their parents were........One morning at the buffet by the pool (the other was fairly crowded and we had spa appts) we had to hold our plates down because the wind was so bad. It rained off and on and the decks were always wet too. When we stopped on Coz, it was actually a very pretty day, temps in the 80's but water temp was around 72. I got in, but it was cold. People had on short on the ship, but be sure to take a sweatshirt and maybe a sweatshirt type hoodie. If you think the lounges are kept cold during the summer, they are really cold in Jan. I still think it is a great time to cruise though.........just different that summer months even though it is Mexico, Bahamas, etc.
One word, COLD....I live in south Florida and January is our coldest month...as Boostermom said, the open decks will be chilly and the Bahamas about the same temps as south Florida, the water will be too cold to swim in unless you're from Michigan...