Ian: The original GOG downloader broke for me today, it wont let me log in despite already being logged in to the website and Galaxy. This is not intended to provoke a Galaxy/downloader debate, just answering the thread question.
I tried to test mine just now to see and it wouldn't connect at first either, but I manually updated my login info and it now appears to have logged in ok. Going to see if it will download something now...
Update: Ok, I just successfully downloaded some small manuals with it (Star Trek 25th) to test it. It's possible the game you tried to download might have broken links or something perhaps as they appear to be having web-rot increasingly over time.
If you're sure you're using the right password, might want to try making a new one perhaps. Oddly though, my Downloader shows a red icon in the systray but green in the taskbar, and downloading is working. It shows "1 NEW" and "1 message" even though there are no new messages, and that wont go away.