When a coal plant fails, it just turns off. When a nuclear plant fails, you've got to get the best people on Earth and tens of billions to contain (not fix) the problem, in one of the most advanced nations on Earth. It made a huge mess that spread to Europe when Chernobyl failed. If nuclear is wide spread, what happens when human incompetence causes a plant failure in a place like Nigeria or Venezuela?

Edit: also, I frequently hear this argument about coal, but it's not a choice between nuclear and coal. Nuclear advocates promote investment in nuclear, but if we were to invest huge amounts into new energy infrastructure, we might as well invest in renewables, rather than another dirty hard to manage source.