I have been advocating for deprecating browsers for a while specially because of security reasons but also from an economic perspective.
Is your server vulnerable to SSL attacks like BEAST and CRIME? Check it out from command line or online.
My recommendation is anyway to upgrade your OS and completely remove support for older browsers. That will give you peace of mind about exploits to vulnerabilities in the front end. It is very common to see really old Linux servers, for which automated package managers can't solve vulnerabilities as the OS version is not longer supported.
Unfortunately support for TLS 1.2 is poor so for now a perfect 100% protection is impossible without banning most major browsers.