We went the last week in January in 2001 and it was in the 50's. We wore jeans with long sleeve shirts and took sweatshirts or hoodies with us. It was colder in the morning. (kids had to wear hoods) Evenings weren't a problem because of the parks closing early, we were back in our hotel by 6 or 7. It did rain one day and was pretty chilly with the rain, but it ended quickly. Of course, it was too cold to swim.
BUT - - the payoff was VERY short lines - the longest was 20 minutes for Dumbo. We've gone in various seasons and January is my favorite.