I wouldn't worry too much. In 35 cruises I have yet to have a real problem with kids. The only real problems I've had are with older people who really need to practice what they preach in regards to behaviour.
I wouldn't disregard the kids programs. I know my granddaughter loved it when we went on Carnival. We still had plenty of time with her but she also enjoyed being with kids her own age and doing some fun "kid" activities.
Take care and enjoy the cruise.