I finally managed to talk with someone at Apple Support about this. After clearing my caches, emptying trash, and rebooting, it was still having the original problem. So what we did was reboot into Safe Mode. Then when I fired up GB it did all the same stuff (downloaded the additional content) and asked for my password. However, this time instead of failing, it actually completed.
Recommendation then is to boot into safe mode to complete the GB downloads.
Reboot in Safe Mode totally "fixed" the no internet connection problem.
Shut down your mac, wait 10 seconds, turn it on, as soon as you hear "the startup chime" , press and hold Shift.
Run GarageBand, try the download.
Download, install (it will ask for you admin psswd), finish installation.
Restart your mac so it can go back to reg mode. Enjoy.
I read part of this frustrating discussion ... and did something different that it seems to be working (I still need to wait a bunch of hours for it to finish downloading).
From a terminal, I changed my current user for the management user
su - managementuser
then, I went to the garageband application directory and under MacOS found the executable file. Executed it.
And when asking for the library it worked.
As it seems that the safe mode works, what it is happening is that GarageBand needs super-user rights to do some tasks, in particular related with Spotlight.
In few words, this is a programming error :-)
I went into my system preferences > network > advanced > DNS and added 184.108.40.206 (I think that's Google, but whatever). Anyway, I essentially bypassed my DNS settings that were assigned by DHCP and suddenly, I was in business. I'm not sure if this was coincidence or not, but after about ten failures in a row, this is what seemed to kick it into gear. Give it a try!