I frequently get asked this question as parents of young children are often looking for a few hours alone while their kid(s) are happily enjoying a safe, supervised children's program.  There is no one answer as each cruise line has different requirements and I will list the minimum ages for each cruise line below. Remember that this is not the minimum age to cruise, just to participate in the children's program!

CARNIVAL CRUISE LINE:  Age 2 is the minimum age for the Camp Carnival program.

NORWEGIAN CRUISE LINE:  Age 2 is the minimum