Yes, the Pro version of FeedDemon supports password-protected feeds. When you subscribe to a feed that requires authentication, FeedDemon will ask you for the user name and password before adding it to your subscriptions.
If at some point the user name or password changes, FeedDemon will ask you to re-enter your login the next time it updates the feed. Or you can change the login at any time by clicking the "Passwords" button in FeedDemon's options.
Important: Google Reader doesn't currently support password-protected feeds, so you can't add them to synchronized folders. Instead, you must add them to folders that aren't synchronized with Google Reader.