Bringing children of any age to Positano is easily done.  Though it may seem like a place that caters mainly to adult taste (gourmet food, architecture and classical music rarely appeal to a 5 year old the same way it appeals to a 35 year old), there are many family-friendly activities that can be done in Positano, still leaving time for mom and dad to sample a more traditional vacation.

As Positano is right on the coast, going to the beach is always an option.  With the beauty of the Mediterranean coast line and the mildness of the weather, spending an afternoon at the beach with a picnic lunch and sand-castle building equipment could be fun for the whole family.  For those who really enjoy the beach, a ferry can be taken to the nearby island of Capri for an even more exotic beach-going experience. 

EDIT - You can't build sandcastles on Positano. The beaches are not sand - they are pebbles.