The solution with paper currency was to put a serial number on each dollar, and specify what mint the money came from. The fix with crypto currency isn't quite as simple. What's to stop me from creating two digital dollars with the same serial number and claiming they came from the same mint?
Crypto currency uses a similar concept to the serial number, but also employs some cryptographic techniques to make sure that these numbers are unique across the currency world.
A detailed explanation for this methodology can be found here.