This problem has a lot of sides, most of which it's hard to see how to address as a concerned layman. The one thing we non-scientists definitely can affect is funding, both by lobbying our political representatives to support funding for reproducibility and, for those of us with the financial means, by directly funding it ourselves. The Center for Open Science is a 501(c)(3) nonprofit organization that, among other things, funds the Reproducibility Initiative. The goal of the initiative is pretty simple: to independently validate significant research results. They're focused primarily on biology at the moment, and there is already work underway to replicate results in cancer cell biology. I'm a donor. This doesn't fix all the systemic problems with science that contribute to the high error rate, but it's at least a concrete step we can take to address one of the problems.