Access issue: what window?

I’m currently reading the design and building oriented sections of Cohen and Rosenzweig’s Digital History, and remembered something I keep forgetting to mention. One of the issues which kept coming up about websites and web-based history interactives when I was with the museum was the fact that some minority populations primarily access the web via mobile phones (and not necessarily smartphones, either). How should/will this impact design of sites which want to do truly “public” history?