My wife and I went to Bali in August with a group of children, their friends and young kids, including two under 1. They were all welcomed at the various villas we stayed at in Seminyak. Baby sitters were $7-$10 per hour and were great.
We have been to Bali many times and Balinese seem to genuinely love young kids.
To add, I just asked my daughter, the mother of one baby. She said $7 was the overtime rate for the baby sitters. It was normally $5. Check out villas.