Exactly how many MB of data do you need to transfer or how big is the hard drive? Once you determine that you can decide if you want to shell out extra money for a new floppy drive, network card, etc.
Since you can't get the old drive to work with your new PC you will have to transfer from old PC to new. Word of caution - transferring files via serial is SLOOOWWW. Depending on the amount of data it probably means leaving both machines on all night while they transfer files. Parallel is what I always used but most new PCs don't come with a parallel port.
The way I see it you're gonna have to plunk down some money and
a) buy a NIC for the older PC and use MSLAN Manager to connect the old to the new via a CAT 5 crossover cable (faster than serial).
b) buy a PCI parallel port adapter for the newer PC and transfer via parallel laplink cable (also faster than serial). I don't recommend buying one since it will probably never be used again once you transfer your files.
If you are having trouble getting interlink to work with DOSBOX try using FreeDOS in MS Virtual PC 2007.
I use USB->Serial adapters all the time, they work great with external modems and terminals, should be no problem using it to transfer files aside from the slow speed.
Before you ask USB->Parallel adapters are the opposite - they only work for printers and nothing else (no transfers, ZIP Drives, scanners, dongles, etc.).
EDIT: If interlink is still giving you problems try this.