Steam doesn't support this. But...
You could use the Windows Scheduler or a program like Watch 4 Folder 2.2 to watch a folder and then run batch file or AutoHotKey script. Unfortunately, I don't have a working piece of code for you so that part (the tricky one) will still have to be puzzled out.
Have you thought of just installing Windows Live Mesh on both computers? It has an remote-across-the-Internet feature that is killer. In fact you can remote desktop into one computer just by signing into your LiveID in Internet Exporter (only supported browser), which is super handy at times. But it's not as easy as just popping a file in DropBox.