When attempting to Add Basemap "An exception occurred during a WebClient request" message appears. Step 5 works from Troubleshooting issues adding ArcGIS Online basemaps to ArcMap | Support Services Blog
So, a basemap can be added, just not with the Add Basemap interface. The above link suggests: "If you can add a service successfully, it indicates a problem specifically with the Add Basemap command. If this is the case, you should consider contacting Esri Technical Support to further troubleshoot the issue."
Any assistance in further troubleshooting the issue would be appreciated.