No, saving the user name in a cookie to flag that the user it authorizes is not the right way, and it's definitely not safe.
It would be quite easy to edit the cookie to be someone else's user name, and voila! you are now authorized as that user.
Instead you should store some information in the cookie that is not specific to the user, but is specific to the user session. You could create a value from the browser string (scrambled in some way to make it less obvious) and store the value both in the cookie and in the session data on the server. When the user sends the next request, you can verify that the value comes from the same session and the same browser configuration.
It's of course not totally safe, as you can spoof both cookie data and browser string, but it's a lot safer than putting the user name in a cookie, and somwhat safer than only relying on the session id.
To get an authentication that is really safe, you have to use SSL.