I visited Bali a few times in the early 2000s in my early 20s and again in 2010. I am sure it has changed again in the last 5 years. in 2010 I didn't feel a need to return.
I am looking at a relatively cheap possible family get away in early Dec -flights to Hawaii are expensive, its too early for Japan, so my thoughts go back to Bali. It does have that constant pull drawing you back I believe. Question is, what are the places to go for families that are no where near/like Kuta and Legian? Has anyone taken the family there and if so, where did you stay that you would recommend?
My thoughts are Ubud, North Coast - beaches and snorkelling, Mount Agung, and obviously a few side trips to surf where possible. Surely there is a Bali there to explore that is away from the tourist traps?
Lots of options away from Kuta. How about Nusa Dua or Sanur? They're offshore at that time of the year too.
If I had kids and was taking them to Bali I'd manage my surf time around the swells and spend the rest of the time along the East Coast, around the north coast, and through the villages in the centre. If your family is into attractions there is plenty of that, but it certainly adds stress to it all - the driving, the hassle etc.. Not having to get in a car every day is a great feeling. So many otherwise great days out can be ruined by sitting in traffic.
Kids seem to froth pretty hard just lurking at the beach - so somewhere like Amed would be nice... Safe and decent snorkelling, not too busy etc. You can do freediving courses there if you're into that kind of thing too.
My other advice would be to find somebody to recommend a driver. A good driver can turn a stressful trip on the tourist circuit into a relaxed, funny trip through parts of the island you might never have thought to visit.