It depends how old your children are, and how late they can stay out without being miserable.
We did Epcot for New Year's Eve this past New Years. LOVED IT! We started the day in Magic Kingdom around 10am with our 3 kids (9, 6, 4) and used our 3 fast passes there, had lunch in the park, and then went back to our Resort. Even though the park had reached a phase 1 closure before we even arrived, we didn't think it was that bad. We were able to walk around easily and get to the rides that we had already reserved and we even had a short wait for lunch at Pinocchio's Village Haus (which was GROSS, for the record). After that we went back to our resort to relax and swim (and I did laundry as we were leaving the next day) and then we had dinner at 6 o'clock at Trail's End in Fort Wilderness. We decided to leave the kids with my mother in law so we could have an adult night out. We didn't even get to Epcot until maybe 8 o'clock and the world showcase wasn't that busy. There were already people set up around the Lagoon for Illuminations, which I thought was crazy. I mean I guess they wanted to see Illuminations, but we weren't going to waste all of our New Year's Eve sitting on the grass. We didn't go to the lagoon until right before the fireworks and everyone just pushed to the front anyway, so "saving a spot" was a waste of time.
I have to say the World Showcase on New Year's Eve was fantastic! Every few countries there was a party, lots of music. My husband has been to Times Square said it was even better than that! There were so many people, but everyone was having a really good time. I wouldn't go there with my children just because I wouldn't be able to really enjoy myself-- it's definitely a more adult environment than the other parks. There was a decent amount of people who were dressed like they were going to a club, however my husband and I were in a regular park clothes . There was alcohol flowing everywhere and this is the first time at Disney I've ever seen really really drunk people stumbling around (no fights or anything) and I've been to food and wine for 4 years in a row.
To my recollection Epcot never experienced any kind of the phased closure, and hasn't in many years. I will note that it took us over 2 hours to get back to our motorhome in Fort Wilderness. The busses were crazy backed up and there was traffic everywhere.
We went to MK on NYE-E to see the fireworks there. I'm not someone who will sit & wait for a "good spot" to watch fireworks, so I sucked it up and purchased the Dessert Party for my family. The desserts were kind of "meh" but it was worth the money to have a seat and a pretty decent view of the fireworks with no standing around -- Note: I've been reading that Disney has changed how they seat for fireworks parties, and you may have to wait in line to be guaranteed a good seat. You'll have to do more research if you chose this option. That evening was insane. It seemed like ALL of MK had people standing shoulder to shoulder.
My kids loved the NYE party on the beach at Fort Wilderness. They had a DJ, games, and gave our noise makers and hats. I think every resort does something, so if you don't want to venture into the parks there's always that option.
We loved Christmas Week/NYE so much that we'll be back again this year from 12/26-1/1! Is it crowded? YES. But it's still awesome!!!
