Modern graphical browsers usually have a small 'key' or 'padlock' symbol at the bottom right or bottom left of the screen. When the 'lock' is closed or the 'key' is whole, the browser is encrypting the information and has exchanged basic authentication information with the other computer.
You can also check whether a browser is encrypting information by checking the URL at the top of the screen - any URL which uses 'https://' instead of 'http://' is using Secure HTTP. Unlike the locks and keys, the presence of 'https' does not mean that all your information is being sent encrypted, merely that it might be.