I've often been thinking the same thing that Thematic Stagnation
argues above. I have a friend who used to work at Jeugdzorg, a Dutch organisation that occupies itself with the supervising of parents who, for whatever reason, have trouble raising their kids. The problems in those families ranged from mentally disabled parents to child abuse. I've heard a fair amount of horror stories and it does make you wonder why just anybody is allowed to become a parent, even after several kids have been taken away because they were maltreated
One of the problems with restricting childbirth is its practical application. For instance, if a license is required for becoming a parent, how do you keep unlicensed women from getting pregnant? You might impose contraception on everybody who does not have a license yet, but this gives huge ethical problems with regards to human rights. Apart from that: who gets to decide who can become a parent? And for what reasons? This starts to smack of eugenics.
Then there would be the idea of requiring parents-to-be to follow a parenting course. You'd first have to decide what good parenting is. Experts don't quite agree on that one, yet (attachment parenting or crying it out, anyone?). Even so, this parenting course is already a more feasible idea, but: how do you determine who is expecting a child? Do you require doctors to tell the government when one of their patients is pregnant? What about expecting women who don't go to a doctor? And, what if people plainly refuse to go to such a course, or what if they fail? Do you put them in jail, where they can't take care of their child at all, or do you take the child away? Wouldn't that be even worse?
As logical as it seems to restrict childbirth, the ethical and practical problems are huge. You might wonder if that's worth it, seeing as most parents I know seem to do well enough.