I still have this same problem, and have had it for many months. I gave up trying to use ArcGIS Explorer, but then build 1700 came out, and I was hoping that build 1700 would solve it. I uninstalled 1500 and installed 1700, with the same result - no basemaps, and I get the message "The basemap contains one or more invalid layers." I opened a thread months ago on this issue. None of the suggestions worked. The suggestion to update video drivers was new, so I tried that, still no basemaps. I can add (some) other data from ArcGIS Online, such as USA State Plane Zones, but none of the basemaps show up. I am at a complete loss.