I honestly think it is simple for some in the US.
If people had something else to remind them of the consequences, there wouldn't be so many fucked up little children around the world. The Chinese practices are a bit much, though.
I would theorize on required sterilization any time the govt needs to come in and support your children, you have children but cannot provide a family, or you take on more than you have ability for in your own means, plus a reasonable passing down of blood through two children to start with. (That is considered adequate by genealogists, and if the parents need social support for those two children, that is permittable - but no more than that.) This might sound like rich families could have more children, and it should be right to a ceiling, as the poor families cannot provide a stable life or income for the children. Their parents are probably either working multiple jobs in order to take care of them, or the children are in a Welfare House, where Jabba the Slut shits out a differently-colord kid every 9 months.*
The fine for bringing a child you cannot support would be for the child to go to adoption and you have to pay a foster charge for that child (unless said child goes to someone else within the family), on top of required sterilization. If you cannot pay, public service. Lineage is still observed through all of this, through foster care.
This, ultimately, is based around the child's welfare and not that of the parents or what they can bring in through Welfare by having more dependents. It also would also tend towards a national genealogy bureau or something similar, and help avoid current Social Security messes in the US as well as create a more concrete system to catch the parents from split families who do not contribute to the upkeep of their child.
* Not a racist remark, just a rather
accurate remark in many occasions. When you're listening to your police buddies reporting a paddy wagon request over the band, using the phrase "four to a whore, and nobody brought the Saran Wrap - notify social services for the children", it paints a pretty good picture.