Eugenics is a sticky area of course. It seems to be a common theme in sci fi books that future earth will require "permits" to have children, dependent on things like genetic health and lifetime success. This has its good and its bad points, though it would certainly be oppressive to our current society.
I think it's outrageous that people can have 10 children on social assistance, though I'm not sure if there's a reasonable way to really prevent this, or even if the majority of people would be horrified by the idea. I would consider voting for a law that restricted number of children to perhaps 2 to avoid excessive population growth (if that were an issue, though in most developed countries it isn't at all). Or perhaps even 1 child, with extra "permits" available for qualified people. Is that sounding a little crazy yet? The drawbacks of such a policy might be worse than the advantages, though.
On a side note, I think people on social assistance should be strongly "encouraged" not to have children temporarily, until they're self-sufficient, since I know from family experience that it's EXTREMELY hard to work your way up to being self-sufficient when you have to pay for childcare (especially for multiple children), with entry-level jobs not paying enough to cover it. anyway, /tangent
The idea of improving your health genetically is certainly attractive, though the implementation is always the sticky part. Anything with such dramatic effects needs to be very well-researched, and even then, we could expect to have some ill effects.