I think children should be allowed in premium cabins, as long as their parents or someone are willing to pay for the seat - why not?

The only problem, as mentioned by many, is when parents are unwilling to get their children to behave in an appropriate manner, or are unwilling to rectify the situation. The worst experience I've had was on a redeye when everyone was trying to sleep (and people probably paid for the J seat just so they could be properly rested), and a baby kept crying every 30 minutes. I blame the parents in that case. They could and should have just brought the baby to the galley or something.