The Federal Reserve bank holds large quantities of currency. The Fed also owns securities and makes loans from which it profits. There is an awful lot of money available to fulfill demand without having to print more currency.
The Fed uses open market operations to do this. If the demand for currency increases (bank run) then the Fed offers currency to banks in exchange for securities (treasury notes, bonds, etc.)
The assumption that the Federal Reserve is also the Treasury is the common error. They are two distinct entities which have different ways to correct the money supply.
That's assuming institutions will buy treasury notes and bonds or will even have the time to do so. I imagine a modern day bank run (at a national level) could happen in a matter of hours, possibly in a matter of minutes. If people can't withdraw their funds on the spot, their confidence in the USD will go from 100% to 0% before the teller is able to finish her apology.
I don't think that value of USD will ever go to zero, because there will always be demand for USD as long as The US exists and demands payment for taxes in USD.