I spent New Year's Eve there a couple of years ago, and am going back this year, but not until the 4th or 5th of January. I think the New Year's Eve experience is probably best for adults or teenagers and fairly miserable for young kids.
However, that doesn't mean you can't have a great trip during that time if you plan carefully. Our kids (and us) had a blast, but we simply bailed out of the park and spent NYE evening at our hotel watching the fireworks peacefully. At the time our kids were 12 and even for them the crowds were unbearable and intimidating on NYE after about noon.
We arrived on the 29th and spent three days at the HoJos and using parkhoppers to avoid crowds as much as possible. I bought the parkhoppers with enhanced fastpass, and during that christmas-new year's week I wouldn't do it any other way. Despite elbow to elbow crowds we were able to get on ride after ride with no more than a 15 or 20 minute wait. When the fast pass lines became ridiculous or the fastpasses sold out by noon, we would go amuse ourselves in DCA or go back to the hotel for dinner and return later at night.
It is no longer possible to get enhanced fast pass options unless you book your trip through AAA and stay at a Disney hotel. If you plan on being in the parks in the days before New Year's Eve, I would definitely take this option. I'm getting my tickets this way even for my trip on the 4th-7th because the 50th anniversary crowds are so large this fall.
Be aware that on New Year's Eve itself, fastpasses are irrelevant. If it rains, the park may be empty and everything available. If the weather is tolerable it gridlocks around noon and people can't really get much done after that. I think the NYE experience at
Disneyland is something very different than the normal visit. If your goal is to just be in the park and celebrate the New Year and see the fireworks from within, it is worth the crowds and the wait. If you're like me and prefer the rides and like fireworks from afar... consider doing all the rides and things in the days before or after and not fighting the crowds on that particular day. We were in Disney early in the morning and rode a few rides, but by noon we went to DCA and enjoyed ourselves over there. By 4 p.m. even DCA was getting packed and we went to let the kids swim and hot tub. We actually watched the beginning of the fireworks from the hot tub.
If you want to know more about the enhanced fastpass system there a couple of tricks to it that really make it worthwhile when the crowds are this big.