I do this sort of thing a lot.
I have found that a 160MB hard drive is probably too old to do the sort of autodetect that most USB-ATA adapters require. These were the days of entering the harddrive parameters in setup...
The best bet for this is to get a PCMCIA network card that has PXE boot capability. Or, a PCMCIA card with a supported Etherboot binary on a floppy disk.
Then boot into a diskless linux setup over the network, and transfer as needed. My oldest net boot image for this is Redhat 9. You might want an even older one, look at Redhat 5 or Slackware 3.3.
This would be most painless because you can just transfer the whole thing over nfs. No messing around with hard drive parameters or matching up new and old hardware. No dealing with windows and dos network drivers beyond just etherboot, which has always worked great for me.
Note that you can do wonders with the old Slackware 3.3 boot disks, boot.i and net.i, maybe pcmcia.i With a PCMCIA network card and the slackware floppies, you may be able to get to an NFS mount in only two or three floppies and no PXE boot. They're also super handy because they'll detect your hardware in that dinosaur and tell you what it is.
If you stay in DOS land you'll have to zip up everything and transfer it with a terminal program, which works but requires lots of space and takes forever.
Also getting networking to work on Windows 3.11 if it wasn't already set up long ago is a big pain and should be avoided.
Best not to mess with the hardware or installed software on it at all. PXE is your friend!!