I have only been on New Year's Day once, January 1, 2008, and it was probably an atypical year because of the weather. It was windy and foggy and cold the night before and I think it rained during the middle of the night. It was not raining on the first though.
We did not stay up real late the night before, so we were at the park for opening at 8 AM on New Year's. It was excellent. I think many people do stay up the night before and then sleep in. I think there are also a lot of other things going on on New Year's Day that attract people's attention, such as the Rose Parade and football. The morning hours were very pleasant, and several CMs told us those crowd levels were normal for New Year's Day. We did not stay all day though, so I do not know how much crowds picked up in the afternoon and evening. Also, like I said, this was 2008, and crowd patterns at Disney have changed a lot in just the last few years.