Your check cashing policy should clearly define ID requirements for check cashing. If it doesn't, it appears you have an opportunity to enhance the procedures.
If a person is authorized to work in the U.S., he/she should have a SSN for work purposes. If the company is filing 1099's, they should be documenting the SSN on those forms. But that is not necessarily your problem.
There could be local statutes impacting the employment of undocumented workers that should be considered as well when determining ID requirements and SAR filings.
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Life without Jesus is like an unsharpened pencil - it has no point.