What you're asking for is not exactly straight forward. Even when connecting two computers with both ends being USB it typically requires a bridge component to facilitate file transfer. 
Likely, you would need some kind of terminal program or something on the DOS machine to tell it to push data out through the RS232 serial cable and another program on the Win7 side to receive it.
You would likely have a much easier time plugging your DOS machine in with ethernet and doing a simple file transfer. (Assuming it has an ethernet port, though I haven't seen too many DOS machines without one)
MS-DOS 3.1 was the first DOS version to support networking and file sharing. To share files you would use the SHARE program.