Josh,
I've had your problem many times. The problem is going to be 'know-how', time, hardware+software equipment
e.g
1) use diskettes by "slicing" the files into 1.44 chunks - this is the easiest, less hassell, less cost method, quickest
2) use USB conenctivity - your windows 3.11 may not have the drivers for this side of it
3) 'lap-link' - I think cross over cable is correct for this one .. but you need the software
4) small ethernet hub - hassel to setup if you don't have the hardware & you need to configure the O/S so have the CD's ready as it could ask you for the drivers & NETBEUI protocol files.
If you can .. Option 1) is easier but if you want to learn .. do option 2) , 3) & 4).
To answer your question : If you have a network card + cross over cable then this will not do as you need a straight through cable and a "ethernet hub" (option 4). So the answer is NO you cannot.
But the cross over cable may help with the 'lap-link' cable - assuming you can connect via the serial ports. Nortons PcAnywhere may also be able to perform the function as the 'lap-link' software