What is claiming that your browser needs to be updated? If it's a web site, most likely the web site is just not written to recognize newer versions of Safari and hence it's the web site that is at fault, not your version of Safari. If you'll post the URL to the site, we can check it and see what results we get.
Go to Safari menu -> About Safari and tell us what version of Safari you are running. Go to http://helpx.adobe.com/flash-player/kb/find-version-flash-player.html to tell what version of Flash you are running.
Go to http://www.java.com/en/download/installed.jsp to check your version of Java. Do not download anything new without checking this user tip to make sure you aren't opening yourself to vulnerabilities: