I would consider Norwegian Cruise Line's latest ships ("Norwegian Star", "Norwegian Dawn") to be a good choice. They offer lots of dining venues (eat where you want, when you want), including a special section for children in the buffet restaurants complete with small tables and chairs. There are also lots of activities for children on these ships.
If the flexible dining times are not that important, any of the big standard-market lines (e.g. Carnvial or Royal Caribbean) would be okay, as well. Those lines offer fixed dinner times with the possibility to have casual dinner at the buffet restaurant during a given time frame.