Basecamp 4.4.6 will export KML files which Google Maps will import.
In Basecamp
File / Export / select route
select Type in drop-down list:: KML 2.2 Document v2 (*.kml)
Save
in Google Maps
Sign In
navigate to MY Maps
Create new map
select Import
browse your computer for file you saved as KML
or drag and drop KML file
it will load your file into Google Maps
be sure to click on Untitled Map and give it a name
it is now a Google Map and you can share it or do whatever Google Maps does
