under what conditions do governments have a responsibility to curb
overpopulation?
Government often have the authority to device policies to regulate population, but almost never have any enforceable legal obligation, or even unenforceable moral obligation, to do so.
Governments are morally expected to further the general well-being of their people, and population levels may figure in to that, but little more.
Also, as more and more countries see their annual births fall below their annual deaths, "overpopulation" is often not what they are worried about.
Concern about overpopulation faded when it became clear that demographic transition from having births at more than the replacement rate to less than the replacement rate, was essentially universal, and quite rapid, in countries that reach a certain threshold of economic development (for reasons that are still not fully understood). The very disruptive effects of China's "one-child policy" also discouraged policy efforts to reduce overpopulation.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/MPJ5h.png)
Countries in green have stable populations (based on TFR, total fertility rate), those in blue have declining populations in the long run based upon current births, and those in red have growing populations in the medium to long run based upon current births.
(Wikipedia image source).
A more direct measure of population growth and decline is here (with the reverse color scheme and including immigration effects):
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/t6GuP.png)