There is no guarantee that any female tenant would not become pregnant and have a child while living in your unit. That is always a possibility. (It is even a possibility that a male tenant could receive custody of a child who was previously living elsewhere.) You could not evict the tenant for a child since that would violate Fair Housing Law that states you cannot discriminate based on familial status (children or marriage). Unless there are more than 2 people per bedroom + 1 (or other occupancy law) in that unit, you have no grounds to issue any violation notice.

The child is doing no more than making a normal amount of noise for any child of that age. It is not excessive, after hours, or in violation of any law. The downstairs tenant needs to accept that it is a fact of life in a multi-unit building and honor their lease. A child's noise is not a legal reason to break a lease. You need to accept it, or be sued by the upstairs tenants.

The best solution is as P-Bone said. Rugs with thick pads will cut the noise.

And for your own sake, stop telling people you do not accept children. Remove that discriminatory statement from any written rules! Don't issue lease violation notices based on it. Or you could lose everything in a lawsuit, including that property! All it would take is one phone call to the Fair Housing Administration, to HUD, or to any legal aide attorney. This case would be cut and dried if you were dumb enough to write that into your rules and issue a violation notice on it.