Though you don't mention the ages of your children, we ourselves sailed with our 9 and 10 year old children last spring. We sailed Carnival Sensation and the kids had a great time. Once I was comfortable enough to let them go to the programs on their own, we all had a great time.
The programs are broken down by age 3-5yr, 6-8yr, 9-11yr, 12-15yr and 16&over. Our ship had 2500 passengers and 800 were kids. You would have never know it because you never saw the kids.
We had so much fun and are heading out on Carnival again this spring. This time the kids are brining friends (9 total between the ages of 10 and 17). They will have a great time.
E-Mail me if you would like further info.
Good luck and either way, enjoy your trip.