There is a workaround if you have the capability to create ArcGIS services. Create an MXD using the appropriate geographic coordinate system (such as WGS84 or NAD83) and set the Units (in the Data Frame Properties dialog window) to Degrees Decimal Minutes. Create the service then bring the service into a NEW ArcGIS Explorer Online map using the 'Advanced' option. You want to add a Service URL. Once you do that your service becomes the base for the map and the coordinate display will be in Degrees Decimal Minutes.
However... doing this means you will not be able to select any of the ArcGIS Online basemap services (the option becomes grayed out). You can add any other available service or import shapefiles, CSV files and GPX files as normal
The whole process is a bit cumbersome, but it ends up working quite well. We actually create fairly complex maps in ArcGIS Explorer Online that are running in our custom local coordinate system, and the application handles the on-the-fly transformation from State Plane to our local system efficiently.