Just smile and say "I'll show you mine if you show me yours."
Seriously though it is becoming a big problem and I don't blame you for being reluctant to give them your DL unless the they give it right back, and don't try to photocopy it or write down the number. I might weigh the fact of what kind of business it is, is it a security risk? Something like a bonded warehouse? Someplace that what they manufacture is at risk of being photographed by the competition? Is it someplace that your company delivers to all the time? Maybe your company and the customer could make some arrangements.
If you are union ask your BA for guidance. Maybe take to the HR department in your company and to the highest management you can get to. I would put it in writing and make sure all the powers to be get a copy. Explain what your fears are and ask them if they were in your shoes would they be willing to hand off there personal ID to some cop want-to-be. What would be the consequences if you asked not to go there? (Asked not refused.) Is it worth quiting/fired over it? Remember that in most states you are considered being an "at will" employee. In other words you are free to leave and they are free to fire you for any reason.
As far as it being legal I doubt if there is any law against it. It is their property and they can deny access to anyone they want. You could also spend a few bucks and ask an attorney but I bet it would be a waste of money.
Good Luck.
Surrendering CDL to customers? 
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