I have Version 10.13.6, They want me to Update my Safari. Do I have to update my Mac to latest version Too?

Sorry but Safari is now deemed part of macOS by Apple and other than security updates Safari is only updated when macOS is. For older versions of macOS like you are running your options are to use a third party browser like Firefox, Opera, Brave, etc. But even some them now require up-to-versions of macOS because of privacy and security concerns, as well as features not available on older macOS versions.

Well, probably OK, the problem is the Internet & security have changed greatly, so things outside your Computer can require demands that your old OS & Older Hardware cannot cope with, but as far as things not requiring working outside of your Computer they should work the same until it dies.

iKrupp's suggestion for a different browser is spot on as far as stretches the Internet usage a bit longer.

Well put! :) 👍