We have:
1. Very nice USB interface chips from www.ftdichip.com and other companies
2. A lot of very reliable and good proven programs round the world working under the DOS
3. New PCs without COM or/and LPT ports
But there are not, unfortunately, drivers under the DOS to support the mentioned chips and simulate the COM or/and LPT ports.
So if one of young programmers accept a challenge and write such a driver, it will be a real feat.
Why you don't have recompile your DOS app on windows?
In my opinion it's could be easier than writing DOS driver for RS232-USB dongle.
How do you realize serial communication in your DOS app? You utilize int14 bios interrupts, or directly read/write to uart hardware?