Why can't I add a base map to my data frame in ArcMap?
I am using ArcGIS 10.2 for Desktop and the options under Add Data for Add Basemap and Add Data From ArcGIS Online are both disabled (grayed out).
Look to the lower left of your computer's Taskbar, you should see a small globe icon. Right-click & "Test Connection Now". You will connect to ArcGIS Online and the ability to add Basemaps is restored.
The unfortunate greyness is from not being auto-connected to ArcGIS Online. In the past just being online was good enough to pull a service, now one must be connected to their ArcGIS Online.
To reproduce the symptoms that you describe (i.e. Add Basemap and Add Data From ArcGIS Online both disabled), I unplugged the internet cable from my laptop and started the ArcMap application of ArcGIS 10.2.2 for Desktop.
The Sign In and ArcGIS Online options under File in the Main Menu were also greyed out.
To restore all of the above options I plugged the internet cable back in, and restarted ArcMap.
Curiously, I had to restart ArcMap twice: