I know the original question is quite old but I'll update my working solution here just incase in case someone came across the question like me.
MacOS Mojave 10.14.2
Basecamp 10.4.0
1. Grab your map from http://garmin.openstreetmap.nl/ - I downloaded the "osm_generic_new_macosx.zip"
2. Unzip the file. You should get a folder "OSM generic routable new <something>.gmap"
3. Use Finder to locate your Application folder.
4. Hold Ctrl- click on the Basecamp app icon. Select "Show Package Contents"
5. "Contents" folder should show up. Select it and you will find a whole lot of files and folders.
6. Drag your "OSM generic routable new <something>.gmap" into "Resources" folder.
7. Start (or Restart) your Basecamp and your OSM map should show up.
Hope this helps!helps!