Our past two trips were in January and overall, they were good trips and the crowds are more manageable. The downside is that last year in particular, it was COLD. We are from the Northeast, so we know cold....and it was only in the 50s for a few of our days there. We were walking around in winter coats the whole time and it was reeeeally cold at night. I would just say that as a warning, because if you are banking on a trip that's warmer than home or you're planning on swimming or walking around in tank tops, just know that there's a chance that that won't happen! We also had a few warm days mixed in (70's) which felt great, so overall, it's just going to be a risk you have to be willing to take!
Splash Mountain has been closed both years while we were there. Other than that, I don't think there were too many other major closures that I can remember. The crowds were decent, but I have to be honest, they were still larger than I thought they would be! We used to go in the off season years ago, and you could truly walk on to rides. I think those days are over! If it's going to be a really low season, Disney seems to find a way to run promotions to be sure that things are filled to capacity anyway.
One thing that was a bit annoying was that towards the end of our trip (so around Jan 11 or so) we noticed a BIG increase in Brazilian and South American tour groups. I can't say that it made things awful, but they were loud, swarmed certain areas and clogged up the works quite a bit at certain times. It was a bit frustrating. Plus, in our resort (we were in Sports), they overwhelmed the rest of the smaller families that were there. So it's just another thing to factor in. On our last day there, a big cheerleading competition was beginning and there were tons of them there too and they were loving the acoustics of the center areas of the rooms. Lol. That was on our very last day, so I don't know what the noise level was at night, but during the day, it was hours of non-stop cheering.
Overall, it's still not a bad time to go. We are going back again this January if that's any answer! I would say just be prepared for the cold in case it dips down. Have fun!!