We're making our second NYE trip in a row - it will be crowded. I'm talking c-r-o-w-d-e-d. After last year, there was no way I was staying anywhere where I had to rely on bus transportation if I wanted to go somewhere in the afternoon on New Year's Eve. The Magic Kingdom was closed based on capacity around noon last year, and traffic was at a standstill.
Depending on your budget and what is available, I recommend an Epcot area resort if you're spending NYE at Epcot or Hollywood Studios, an MK resort if you'll try Magic Kingdom or Epcot, and AKL if you want to visit Animal Kingdom park that day.
The parks are manageable with a good plan. If you are really serious about going, book as soon as you can - we arrive on 12/30 and our fast pass booking opens on Halloween - you need to get passes done for *at least* the park you're planning to visit on NYE, and I also suggest booking dining right now in your park of choice if you want a table service meal that day, even if you haven't booked the trip yet. That way you have it just in case (but don't forget to cancel it if you end up changing your mind so you aren't charged the no show fee).
You CAN successfully plan a trip right now for the new year, but you can't be too picky about anything. Pack your patience and be prepared, and read everything you can about NYE at Walt Disney World. This is not a time of year I'd recommend for first timers, so if you've never been before I'd maybe think about a different time to visit.
Because you're traveling with children, please make sure you set up a plan for what to do if someone gets lost or separated in the crowds. In certain areas the crowds are intense, wall to wall people, so to save everyone some stress, create the plan!
Good luck.