You could start by adopting the system we have in Denmark, which makes the acknowledged father of a child responsible for contributing to the child's upkeep until he or she is 18.
This applies whether or not the parents of the said child have married or not.
The municipal authorities pay the monthly childcare to the mother and are responsible for collecting the same amount from the father. If he is unwilling to pay, the Income tax steps in and adds the amount to his tax contributions.
If parents divorce and the child (children) live permanently with their father, the mother pays child support in the same way.
This works, although it does sometimes have comical effects, as when my sister divorced her husband and their eldest child remained with her mother, while the younger sister lived with her father and my sister and her ex had to send each other the exact same amount each month!
This prevents men not supporting their families. If the same man has more than ten children under 18, the state takes over child support for subsequent children and presumably sends the man a leaflet on family planning.
CUTTING OFF A ROSE BRANCH IN NEXT DOOR'S GARDEN