The solution is even simpler than that: encourage poor people to have one or zero children. Cash handouts to encourage poor men to receive vasectomies for example.
Yikes.
There is a better alternative: it is very well known/studied that women that have a strong position and are educated have less children.
One (poor; your words) man without a vasectomy can knock up many women. Hence I think it's better to focus on the women.
It should be pretty clear why improving education is not a yikes while eugenics is.
I am suggesting that the attempt of suppressing reproduction within a particular demographic, no matter the incentive or perceived "good" that results (cash or education), is walking a very fine line.
Coercion is waaaay less, to the point that I'd like to say educating is not coercing. Education can be undone, unused, forgotten. Vasectomies not so much. I dont think "totally" suppressing reproduction is what happens: those women still want/can/will have babies, just not so many.
The sequel to borat was basically about borat lying to his daughter about what women can and should do and her seeing through the lies eventually. Some countries are really behind the times and "backwards" ideas that were common in Europe 200 years ago can still be found there today.
Why poor men?
You might find this article relevant to your interests[1].
Experts estimate 15 to 40 percent of men undergoing vasectomy experience a sperm granuloma. In some instances, a man may have to have the granuloma surgically removed.