If we're willing to see rates of reproduction as a kind of public good, then why not try to replicate the conditions or yore in which children redounded to their parents' bottom line. Pass a law that gives parents a claim on some percentage of their children's future earnings. This will not only increase the quantity of children produced, but will increase the quality of children as parents make targeted invetsments in their childrens' human capital in order to maximize their take from their kids' future production.
Doesn't this make grown children slaves to their parents? Well, it's not exactly clear how this differs in principle from the way taxes make us slaves to the state. If it turned out that the incentive for parents to produce kids and invest in their skills led to average after-parent-tax earnings greater than average earnings in our current system, and it produced a higher rates of economic growth through greater labor productivity, a stable level of population growth, cultural continuity, and non-collapsing pension systems, wouldn't that be enough to justify it to the grown kids themselves? Shouldn't we want the state to require that we pay the tax to parents, rather than to the state, if paying taxes to our parents produces all these externalities?
Maybe you're uncomfortable about treating people as mere means even to the end of their own welfare. Then maybe you're against taxation altogether, which is absurd.