Earlier quoted context omitted.
That's an exaggeration, sterilisation should be sufficient. Edit: I'm being sarcastic.
Not a bad idea. What if the government offered free tubal ligation and vasectomy. If you later want to have kids that cost is on you to reverse it.
I think there may also be mechanisms for women to get the contraceptive pill gratis if they are on government support.
The issue is though that this requires some degree of responsibility and planning, which is almost by definition missing in people who breed irresponsibly. So you have a lot of people who want a lot of children or are at least indifferent to having them, however they have no financial means to support that many children. I guess if you have not been successful career-wise then having a large family might give one a sense of accomplishment.
The upshot of that is that it's not fair to let these kids starve so the government has to offer welfare to these families. For example it is much easier to get government subsidised housing if you have children. So you end up with a situation like we have in the UK where people sometimes have children because they will get a free house (and if you have more kids you get a bigger house) where otherwise they might be homeless or forced to live with parents etc.
So quite the vicious cycle.