I would prefer to work with my GBA and Mac OS 10.6 Snow Leopard, but I've also got a GBC and a Windows XP machine hanging around.
Hands down the easiest way that I have found is to get a Nintendo DS that has the Game Boy Advance slot (original and DS Lite), and a slot-1 flash cart.
After that it's just a matter of downloading some software. Linfox is great for that.
NOTE: I would stay away from the M3 solutions for this task because one of the better backup programs, written by Rudolph, was updated to single out the M3 and refuse to backup the saved game.
Alternatively, there are (or at least were) solutions –
– that will plug right into a computer. However, this is a much more technical, difficult process.
As I discovered recently, This can be easily accomplished with the Retron 5 Gaming System:
As quoted above, You can then also copy saves to/from the Retron via the SD card slot.
This will work for all Game Boy, Game Boy Color and Game Boy Advance cartridges, as well as NES, Famicom, SNES, Super Famicom, Master System (via Power Base Converter) and Mega Drive (Genesis) carts that support a 'save' feature.