I found the solution at least for me. 
From one day to another Brave didn’t download any more files.
When I clicked on a link the folder selection window opened, where the files should be saved and that was it.
No saving process at all and finally no file 
.
I found several discussions on the net as well over here in the community but they all remained with open questions and had been closed by the system after 30 days without any final solution.
However I figured out, that one of the extensions was the reason for my blockade - In my specific case it was the jdownloader extension - once deactivated everything with the brave download worked well again - So I recommend if you might have similar problems do the easiest things first, deactivate first all of your extensions and see if it works again and then try to find out who is your issue extension.
Hope at least this will work for some of you, I know it’s really frustrating.
Cheers.