Okay so a minimal solution
Physically tie off (reversibly - even if we can't reliably do this now it won't be more than 20 years coming) everyone as or before they reach sexual maturity.
In order to get it reversed you must:
pass a psychological assessment (I know full well I would have failed this part)
pass a theory test to demonstrate you understand the nature of a childs needs both physical mental and emotional.
pass a physical assessment to demonstrate that (with any adaptations/ support necessary to overcome disability) you can look after a child.
deposit the minimum cost of raising a child to maturity for progressive release as the child requires the resources.