Posted for information, for the next person wanting to connect to an older etrex.
It seems that Basecamp for Windows 10 can't communicate with eTrex H handset, and I couldn't find any setting to fiddle with data transfer parameters.
Luckilly I found public source GPSBabel that was able to up and download from the eTrex H without any problems. It had immediately detected the com7 port that the device was attached to and did the job.
There doesn't seem to be much support or help information about the limitations of Basecamp.
Along the way I re-discovered that Comm to USB cords can be problematic. In the past I had purchased a dud cord, and despite there being no logic to the action subsequently bought a different brand of the same sort of cord. Luckilly the second one worked. Fast forward a few years to this experience, and my friend's GPS wasn't showing up as a Comm port. I found my old cord, swapped it for the one that came with the friend's device, and up popped the Comm port and I could talk to the GPS.