It is not easy to accomplish what you are trying to do. The server could be using one of two types of authentication with the client (browser)
1) Transport authentication
2) Forms based auth.
Transport auth: In this case, the authentication is done using the transport connection itself - here it will use custom HTTP headers with a multiway handshake to do the auth.
Forms based auth:This is the traditional auth that is done when you enter your credentials into a form.
In either case, the authentication might have already happened by the time your managed control is instantiated. As someone suggested you can steal the browsers cookies for that domain and use it with webclient but then it may or may not work the way you expect.
If all you want to do is download something, then I would see if I could use the browsers facilities for eg, XmlHttpRequest or some other ajax mechanism to download what you want as part of the DOM of the page that is hosting the control. Then you could read that content from your control, or you could also have the browser inject that content into your control by way of Javascript calling a method/property on your control.
[EDIT]
Find out (using Firebug plugin in firefox) exactly how the forms based auth is being done in the browser. Then you can write code to do exactly the same request/response that the browser is doing. To the website, the client will appear like any other browser based client. Then you should be able to download anything you want from the website.
Hope this helps.