My browser is 10.12 and I'm loosing ability to access my email and some websites. Can I upgrade my server?

Safari is updated when you update/upgrade your macOS.

The model year of a MacBook Air is available at Apple menu > About This Mac.

A late 2010 through mid 2011 MacBook Air can be upgraded to High Sierra 10.13.6 but no later version. See: How to download and install macOS - Apple Support

A mid 2012 MacBook Air can be upgraded to Catalina 10.15.7 but no later. See: How to download and install macOS - Apple Support

A mid 2013 and early 2014 MacBook Air can be upgraded to the latest version of Big Sur macOS 11 but no later. See: How to download and install macOS - Apple Support

An early 2015 and a 2017 MacBook Air can be upgraded to the latest version of Monterey macOS 12. See: How to download and install macOS - Apple Support

A 2018 or later MacBook Air can be upgraded to the current version of Ventura macOS 13. See: How to upgrade to macOS Ventura - Apple Support

Be aware that if you upgrade to Catalina 10.15 or later you will also need to update any non-Apple 32-bit apps. See: https://www.stclairsoft.com/Go64/

or

 https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html

If you can't or don't want to upgrade your macOS, for security reasons you should consider using a 3rd party internet browser that supports their current version with your older macOS. One example is Chrome if you have macOS 10.13 or higher. See: https://www.google.com/chrome/

FWIW, Firefox 115.02 will still run in macOS Sierra 10.12.

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