What version of Win 10? Are you a Windows Insider, and, if so, fast or slow ring?
What do the logs show?
I think I would look at the video drivers and check to be certain that the driver presently installed is in fact Open GL 1.3 compatible. There have been many complaints from Win 10 users that during an upgrade Win 10 "throws away" previously installed video drivers from the computer manufacturer and then reinstalls downlevel drivers from Microsoft. It hasn't happened in my case so I have no actual first hand experience with that problem.
I run Win 10 Pro in a VirtualBox virtual machine on my iMac. The VB drivers supplied by Oracle do not meet the specs but I am able to run Basecamp in 2D mode only. You might try starting Basecamp in 2D mode. (Rats, I neglected to write down how to do that. I'll go start it up, see how its done, and then edit this)
[Edit] Looks like they start it like this:
"C:\Program Files (x86)\Garmin\BaseCamp\BaseCamp.exe" /Disable3D
You need the quote marks because of the spaces in the file names
/End EDIT
Maybe your problem is similar/same to one reported some time ago, with this result:
https://forums.garmin.com/showthread.php?220236-basecamp-crash-directly-after-start&p=610723#post610723
System.Windows.Forms.ni.dll is a file having to do with .NET
When I loaded BaseCamp on my system for some reason it decided to find and download .NET Framework 3.5 even though the BaseCamp specs called for a higher level (.NET Framework 4.6). I think it did so to satisfy one of the "helper" programs included with BaseCamp. Here is a link to a Microsoft Developer Net article which explains how to find out the .NET Framework level installations existing on your machine.
Looks like all Win 10 come with at least .NET Framework 4.6 or above.
https://msdn.microsoft.com/en-us/library/hh925568%28v=vs.110%29.aspx
Otherwise I'd explore the idea in the link Tranquilo2 posted up...