Gmail message about safari
Safari is a integral part of the Operating sytem.

Your version 5.1.10 is the last version available in Snow leopard (your operating system)

Now you have 2 choices:

1- Upgrade to next version of Operating system with new version of Safari (big job)

2- Switch to Google Chrome or Firefox browsers (very easy to do)

I personally prefer Firefox and use it to access my gmail.

Just click on one of the links in my answer and follow the instructions to Install.