To have an idea of the risks of using an outdated browser, have a look through the following for vulnerabilities subsequently patched in Safari and WebKit (the underlying Safari rendering engine) since the last 5.1.1 release. I still run Mountain Lion, no longer supported by Apple since near the end of 2015 (much more recent than Snow Leopard) and I wouldn't dream of using Safari 6.2.8 there any longer--in fact, I've completely removed it from the Dock.
Apple security updates - Apple Support
Apple security updates (2014) - Apple Support
Apple security updates (2013) - Apple Support
Apple security updates (2011 to 2012) - Apple Support
All you'd have to do is hit a site that has been hacked and it might be game over. You'd never know it, except when you find out at some point that your banking and credit card, or other sensitive data, has been lifted. Many of these exploits describe your machine being compromised and taken over completely.
You could, if you like, scan for known Mac trojans with https://www.malwarebytes.org/antimalware/mac/ but getting back a negative will prove little, since a browser exploit that proceeded by way of one of these vulnerabilities might not install a trojan. You may not have been hit with anything, yet, but best to just stop using Safari immediately.
In addition to the risks of using a browser that hasn't been patched for vulnerabilities in many years, there are these two major exploits for Snow Leopard which Apple patched in 10.8 and above, but never patched for Snow.
http://www.zdnet.com/article/major-ntp-security-holes-appears-and-are-being-expl oited/
http://www.zdnet.com/article/first-attacks-using-shellshock-bash-bug-discovered/