The weather in January on the Islands is nice. If you are departing form Miami you might get some cool weather. We left on the 7th f Jan this year and we were freezing at the sail away party on deck. The day after at sea it was a bit windy but much nicer. Once we arrived at our first port which was San Jan it was beautiful. If you are from up north like we are winter vacations are a blessing.
First of all you need to understand "our" Kuki's sense of humor! (Moose's (or is that "meese"?) don't have much of a sense of humor!)!
If you'll post your sailing date and ship it will be easier to tell you about the weather. Several lines post the weather to expect while you're cruising (i.e., Princess and Carnival).
If I were going to "guess", I'd say that January in the Eastern Caribbean would be in the high 80's in the daytime, and low 70's at night.
I agree with suebecdan, bring a light jacket, just in case!
"A truly happy person is one who can enjoy the scenery on a detour."
Its really hard to be specific-- on average it might be a bit on the cooler side when you sail away, the closer you get to the Caribbean the warmer it will become. Chances are your islands will be nice and warm, warm enough to swim and beach. We have run into really hot and muggy weather on some islands. Pack for summer days, but do pack a lightweight jacket, or sweater and lightweight casual pants.