You should be able to connect a USB to serial cable. On the sending computer look in device manager for the COM# of the device.
For example lets say it's COM2. Then run the following two commands from a command prompt.
MODE COM2: BAUD=b PARITY=p DATA=d this command you should only need to issue once with in the command prompt and you will need to replace b,p,and d with the appropriate values configured on the DOS machine
Then to copy you would simply type copy somefile.txt \\.\COM2
For binary file copying COPY somefile.zip /B COM2 /B