How to access serial terminal through USBASP?
From the USBASP documentation:


  
  Planned: serial interface to target (e.g. for debugging).
  


..and further on..


  Firmware:
  The firmware dosn't support USB Suspend Mode. A bidirectional serial
  interface to slave exists in hardware but the firmware doesn't support it yet.


Keep in mind, that's from 2011, so it's possible someone has built an updated version of the firmware with that functionality. You'd have to do some searching (I'm not familiar with any such firmware).

For now, your best bet is just getting a TTL serial-USB converter (they're cheap) and connecting it directly to the Pro Mini.