Bug: Adding a basemap changes the coordinate system of the data frame to Web Mercator (auxiliary sphere)
When adding a basemap to ArcMap 10.1 using the Add Basemap dialog, the map display is not refreshed and the coordinate system of the data frame is changed to WGS 1984 Web Mercator (auxiliary sphere).
WGS 1984 Web Mercator (auxiliary sphere) is the spatial reference of the basemap layers available in the Add Basemap dialog. This bug causes the data frame to change to this spatial reference when adding layers using this dialog.
Go to the View menu > Data Frame Properties > Coordinate System and change the coordinate system back to the correct spatial reference.
If working with the same basemap layer frequently, the basemap layer can be saved as a layer file. In the Table of Contents, in either the Drawing Order view or the Visibility view, right-click the basemap layer > Save As Layer File. The next time this basemap layer is needed, browse to the saved layer file (.lyr) and add it to the map. This method of adding the basemap does not change the data frame coordinate system.