I think most people agree that not everyone should have kids, the problem comes with coming up with criteria, genocide is unfortunately far too tempting to ever give that power to any individual or organization imo. People are better at policing it than you think though, so long as they have easy, stigma free access to the tools to control it themselves.
While we can entertain the idea that some people shouldn't breed(genetic defects that lower the quality of life for the individual or society) or shouldn't raise children(stupid raises stupid), there's no neutral authority that can make and enforce judgments. Even if we make an AI to do it, it would just reflect the view points of the people who made the AI or the sample set that was fed to the AI.
The best we can do is make a society strong enough to cope with these issues, either new and better genetic treatments or more robust public schools systems.