We have done 3 cruises to the Bahamas and Caribbean islands in January. The weather of Bahamas and Caribbean in January will be perfect for traveling -- sunny, warm, and dry. In general, Bahamas will be a few degrees cooler (both ambient & water temperatures) than the Caribbean islands.
For us (from SF Bay Area), Bahama's ocean water was quite cold -- not intolerable, but definitely uncomfortable initially and can only stay in for maybe 30 minutes at a time. However, the Caribbean water will be quite a bit warmer (only a bit chilly for us). Once you get over the initial shock (because the ambient temperature was so warm), you can stay in it for long durations. The warmest water will the Mickey's pool on the ship, especially the ears section, which have black bottoms and after soaking up the sun all day will be like bath water in the afternoon. My DS3 & DD1 (age at the last January cruise) both loved it!!