You mention that you want something open source. I'm guessing that you're probably looking for "no cost" more than "I want to look at the source code".
If you're willing to spend a little bit of money you can get LapLink V for DOS, which is still a shipping product (at $49.95). With a parallel cable (they'll sell you one of those, too, if you don't have one) you can move roughly 5MB / hour. That's better than you'll be able to do with serial (maxing out there, assuming the UARTs on the laptops can handle 115.2Kbps, at roughly 675KB / hour), though LapLink can use serial connections if you so choose. I've used LapLink in these kinds of situations before and it's worked well. 2GB is a lot of data to move over a serial or parallel cable. Yeeouch... 17 days at full parallel speed.
There's a cheaper competitor to LapLink out there, FileVan, but I've never used it personally and can't say anything about it.
If these machines have PCMCIA card slots you might do better to get a PCMCIA Ethernet card with DOS NDIS2 drivers and use the Microsoft DOS networking client to transfer the files. You'll have protocol issues if you try to access a newer server OS with the DOS client. If I were you, I'd probably put Windows NT 4.0 up in a virtual machine on a PC, get it talking to my local Ethernet network alright and pullng DHCP, get the DOS client setup to pull DHCP, then connect the DOS client to the VM. You'll need NetBIOS name resolution for the DOS client to work right, and NT 4.0 will happily broadcast its name. NT 4.0 also won't do SMB signing and will be happy to talk to a DOS client. The only money you'd spend for this solution would be for the PCMCIA Ethernet card (which you might already have) and your time to setup whatever server the client will talk to.