There is a program that does that over a TCP/IP connection.
http://aminet.net/package/util/sys/clipo_v0.1beta
Thanks, but I know that one

Why use the serial port if you have an Ethernet adapter?
Why not? I only use my network for file transfers, which means it's off most of the time. The serial connection is always there.
Anyway, I've written my own utility for this in assembly language. It uses the clipboard handling program 'Clip' from aminet, and all it really does is receive a file from the pc side and output it to the console. Both programs are tied together using a script.
All in all quite simple, except for the XON/XOFF headache (not a problem for ASCII text files, but a problem for binary data, such as file sizes). Turns out that when using XON/XOFF handshaking, two characters (17 and 19) aren't available for sending data. Easy to handle, but when you don't know about it, it's a real pain.