MacBook Air (Mid 2013) which update after Big Sur?

Big Sur is the highest supported upgrade for a mid 2013 MacBook Air. See the following from Identify your MacBook Air model - Apple Support

MacBook Air (13-inch, Mid 2013)

Year introduced: 2013

Model Identifier: MacBookAir6,2

Part Numbers: MD760xx/A, MD761xx/A

Newest compatible operating system: macOS Big Sur

You may want to see if you can install a browser that is more current since Safari cannot be upgraded. See: Browsers by Mac operating system - Apple Community

No newer OS upgrades are available for that model.

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The last security release for Big Sur was 11.7.10 on Sept 11, 2023. See: Apple security releases - Apple Support

There have been other security releases since then that did not update Big Sur as shown in that document.

Using a current browser as described in my previous reply will provide more protection than running an older version of Safari but doesn't provide protection against everything that a current OS (Sonoma, Ventura, or Monterey) would. Apple provides security releases for the most current 3 maxOS versions.