I manage access to a U.S. military base. Can we use E-Verify for badging or to check whether people are allowed on base? Or, could we ask them to show a printout of their E-Verify Employment Authorization as a prerequisite for base access?

No, E-Verify can only be used for employment authorization confirmation purposes.  You are not authorized to use E-Verify for military badging or base access. E-Verify use is limited to employers and other participants who are confirming identity and employment authorization for their new hires (as well as current employees for certain federal contractors). Additionally, regarding whether to request a printout, you are not permitted to ask for a printout showing E-Verify Employment Authorization.