High intelligence does seem to negatively correlated with fertility rate -- one study here, but there are plenty in Google Scholar. http://www.sciencedirect.com/science/article/pii/S0160289607...

But "intelligence" / "IQ tests" are tangled up with a lot of correlating factors and biases so it's hard to say what this means. For instance, simply being poor affects cognitive ability considerably -- https://www.princeton.edu/news/2013/08/29/poor-concentration...

We also know that low fertility is also associated with high education levels -- http://www.jstor.org/stable/2137845 -- and income -- https://papers.ssrn.com/sol3/papers.cfm?abstract_id=838227

I think one key here is, now that infant mortality / childhood diseases are much lower than it used to be, a much larger portion of families really want to use family planning methods to keep family sizes low, globally. This is not necessarily a bad thing. http://www.economist.com/node/14743589

So I would take a statement like "the stupids are outbreeding the smarts" as more that, for multiple reasons, access and knowledge of family planning is not as good for those with less education or income. So rather than making this a moan about dysgenics and Idiocracy, it seems a call to action to absolutely ensure these people at least have the access to the health care they need. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2835625/

Why I fully support every person having access to family planing methods as I don’t think anyone should be having children they don’t want, the relative cost of children would still cause differential changes to the population even if 100% of people had health care access.

Children are much more expensive to the highly educated, both directly through things like housing and education, and indirectly via opportunity costs. The more expensive you make something the less of it will be consumed unless it is a Veblen good.