My wife and I have been 3 of the last 4 Januarys. The second week of January has a pretty low crowd, so if you're looking to do lots of rides, it's the perfect time to do it. The Walt Disney World Marathon weekend is from the 10th to the 13th, so there will be more people then, but still not too terrible (and they're mostly morning people, so evenings are not too bad).
January is the coldest month of the year, with highs around 72 and lows around 50. That being said, sometimes it is considerably colder, and on one of the days we were there, the high was 54 and the park was full of people wearing hats, mittens, scarves, etc... Needless to say, water parks are definitely not a big attraction, and Blizzard Beach will be closed from January 6th-March 16th. I don't see anything about Splash Mountain, but it's been closed two of the three times I've been.