I think Windows Media encoder has the ability to require a username and password to access a stream. It's also free and fairly easy to set up. If you're running a firewall and/or a router be aware you'll probably have to fiddle around with those too before anyone will be able to access your stream.
If bandwidth usage is the reason you want it to be private, another alternative is to host your feed on Gordon's excellent icecast server. That's what I use for my feed and it's been fantastic - he handles the bandwidth so that if the whole RR community suddenly decides to tune in to my feed my internet access won't slow to an absolute crawl. Hosting a feed on his server also allows people to listen with either Winamp, Mediaplayer, RealAudio or a flash based player that requires no installed media player at all. You also then won't have to worry about IP addresses changing or your ISP dinging you for hosting a server on a residential account. I highly recommend it. Click the link in my sig for details. We're always looking for addiitonal AZ feeds.
-AZ