I think someone else already mentioned Seminyak, that's where I'd recommend. We stayed in a great villa there, it's quieter than Kuta but not too quiet - no revolting 20 year old Aussies or middle-aged American or European man-boob types, and some very nice Balinese restaurants. Close to Kuta if you know the area, but without the horrors. Just a 20 minute cab ride (or a 10 minute walk
) away.
If I can find the name of the villa we stayed at I'll PM a link to you.