A lot of this depends on how your HttpHandlers are being used. If they are used within a large ASP.NET app, you could just check whether the user is currently authenticated. If you were using Session, you could check the context.Session["some unique user key"] to see if it is populated and valid.
If other applications are going to use your handlers, then you'll probably want to purchase an SSL cert and include the username/password in the message. They would be encrypted as they are transmitted over the network.