In the U.S.: you can renounce citizenship (which must be done abroad), and then have your child abroad. This should prevent the child from obtaining U.S. nationality through you, although they could still obtain it through the other parent (if any). However, I think renunication by people with prior U.S. residence is fairly uncommon—unless the renouncing person had another nationality, they would become stateless, which is not something you want to be!
Parents cannot, however, renounce the U.S. citizenship of their (born) minor children.
In the U.S.: you can renounce citizenship (which must be done abroad), and then have your child abroad. This should prevent the child from obtaining U.S. nationality through you, although they could still obtain it through the other parent (if any). However, I think renunication by people with prior U.S. residence is fairly uncommon—unless the renouncing person had another nationality, they would become stateless, which is not something you want to be!
Parents cannot, however, renounce the U.S. citizenship of their (born) minor children.