I’m using Mac Mail which from what I can tell does allow username and password to be asked for. The Restricted Site Access plugin generates a page for the */feed/ but it is a login page so the feed reader doesn’t recognize the format since it’s just a web page.
I tried Feed Key which appears to require Members Only plugin to work. That, while it had a few things that were a little clumsy did indeed work. The only problem was that it was an all or nothing for the entire site and had no ability to have the top level open and only certain subdirectories restricted.
I’ll try the plugin David recommended next and see what happens.
I had another thought after getting a little sleep last night. Perhaps I can just use htaccess approach, but I don’t want the users to have to have one more password. So the trick there will be if I can find a way to get the normal user level approach to work with that.