My recommendation for New Years Eve... Pick whatever park you wish to spend New Years Eve at, get there in the morning, and STAY there.
Typically speaking, Magic Kingdom will be the first to fill up. Then Epcot. The Disney-MGM Studios might hit capacity, but it usually just comes close.
When a park nears capacity, there are various levels of closing.
First step is to close the parking lot (this usually occurs only at the Magic Kingdom/TTC). Guests are then re-routed to park at Epcot, and to take the Monorail to the TTC, then to the MK.
Second step is to close off ticket sales. You must have already purchased a ticket to enter the park. No new ticket sales for today.
Third step is to close the park to day guests. Only Disney Resort Guests will be allowed into the park. This is the most common stage.
The final step is to close off the park to EVERYONE. This means the park is FULL.
A few things to note...
It's not a one-for-one thing. So, if 5 people leave, you still may not enter.
Once you leave... you are not guaranteed re-entry. So, don't forget anything in the car -- you might not be able to get back in.
"Capacity" is not a set number. Rather, it is a decision based on many different factors, made by the parks highest management team.