How do I upgrade my browser from safari 5.1.10

Open the Mac App Store and try downloading Yosemite. If you get told it's incompatible or otherwise desired, choose About this Mac from the Apple menu, check if the computer has at least a Xeon or Core 2 Duo(not Core Duo) CPU and 2GB of RAM, and if it does, click here and order a download code for Lion 10.7; the online Apple Store also sells download codes for Mountain Lion 10.8, but it has the same system requirements as Yosemite.

Back up your data and check your applications for compatibility before upgrading. In particular, Mac OS X 10.7 and newer don't support PowerPC programs such as versions of Microsoft Office prior to 2008.

If you’d prefer to or need to stay on 10.6.8, switch to a product such as Firefox.

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The only way to update Safari is to update your OS.

Depending on the exact model of your Mac Pro will be what Os version you can update to.

At the very least OSx Lion 10.7.x

http://www.apple.com/shop/product/D6106Z/A/os-x-lion

Your other option is to download a 3rd party browser like Firefox and use that instead.

With that said, I'll request this be moved to a more appropriate forum since you are running OS 10.6.8 and not any of the classic Mac Oses that the forum you posted in is devoted to.