Safari updates are linked to OSX levels. 10.6.8 cannot run the latest version, Safari 9. Your options are:
1) Update the Mac OS on your computer. Yours will support the current version El Capitan if you have at least 4GB RAM. However, older programs like AppleWorks and any version of MS Office:Mac older than Office 2008 will stop working once you leave OS 10.6
2) If you do not wish to upgrade your OS, switch browsers. FireFox is good for the moment but there are posts that it will soon not work on any Mac OS older than 10.9 Mavericks.
I've been testing Seamonkey which appears to retain OS 10.6 support and it looks and feels just like FireFox. Quite nice.
Don't get either browser from a download hosting site. Get only from their respective developers:
FireFox: https://www.mozilla.org/en-US/firefox/products/
SeaMonkey: http://www.seamonkey-project.org/