There are many examples, from the ancient world to Nazi Germany, of attempts to protect or to increase the birth rate and hence population size. Slides to accompany Prof David Coleman's talk on birth control from governments.
Few can be shown to be successful. France, since 1939, is an exception. The Nazi example put population awareness off the agenda in the West but the persistence of low fertility, with the population ageing which it causes have brought population concerns to the fore in many countries: Italy, Germany , Russia, Korea , Japan. The relatively high fertility of some countries (Scandinavia, UK) is unintended by policy, and probably owes much more to consistent cultural preferences and welfare policies unconnected with demographic aims. Cash incentives probably just bring forward births already intended. Relatively high fertility seems to depend upon a consistent policy of family support, or the ability to make arrangements through the private sector as in the US, on which parents can depend, and a well developed gender equity in which burdens are shared between the parents.