I am running 10.6.8. I now have Safari 5.1.10. What is the newest version of Safari that I can run?

Without upgrading the computer’s OS, the one you have.

If you do, either 6.1.6 or 8.0.5 depending on your Mac Pro model.

(127155)

You're already running the latest. You need to upgrade to a newer, modern version of OS X to be able to run a newer Safari version.

If you have reasons not to upgrade your OS, you can use FireFox. It still supports OS 10.6 and is fully updated and compatible with most web sites.

https://www.mozilla.org/en-US/firefox/new/

IF upgrading the OS is not a concern, we need to know which Mac Pro variant you have before we can recommend an upgrade strategy. Those made before 2008 are hardware-constrained to running no higher OS than OS 10.7 Lion.

Please do "About this Mac" from your Apple menu and, in the resulting window, click the "More Info..." button to launch System Profiler. In Profiler's opening screen you will see this:

Note the red arrow I added; please post the code your computer shows in that line and we can advise you if it is possible to update your Mac to a higher Mac OS level. Also post how much RAM ("Memory" in the above image") is shown. Newer version of the OS may require more RAM than you Mac had when new. For best results, OS 10.7 and higher do better with 4GB RAM; Yosemite can want 8GB RAM depending on how you use your computer.

look here

http://www.macupdate.com/app/mac/15675/apple-safari