The government has an E-Verify program. Does all the work for you.
E-Verify apparently allows a participating employer to verify the authenticity of a Social Security Number.
I've never used the system, but I suspect that as long as the fake documents use a valid SSN and the person uses the name that matches the SSN, the E-Verify system won't catch them.
If that is how it works it makes it a little more difficult to forge documents, but that's about it.
The system is also voluntary. I don't honestly know what hoops the government has employers jump through to use it, so I don't know if employers have a legitimate reason not to want to use the system or not.
However, unless the use of such a system is mandatory, employers really can't be held accountable for not using it. The solution to that is make a decent system that is as easy to use as possible and as reliable as possible and then require its use.