No, you don't need to do anything to enable this. AirPort Extreme by default has all 3 of its LAN Ethernet ports set to autonegotiate 10/100/1000, duplex, flow-control, and auto-MDI-X (auto-crossover). I don't think there's even a way to override that to force a particular speed/duplex setting on a particular LAN port (unless you have one of the ancient UFO-shaped 802.11g models that were sold from 2003-2006 and had only a single LAN port).
You can set speed and duplex (or even "Ethernet disabled") for the WAN port on any of the models, but it's enabled and set to autonegotiate by default, and besides, you've probably got that port plugged into your upstream Internet connection.
Does the AirPort Extreme show a link light for the Ethernet port you're plugged into? (The link lights are built into the ports connectors)
When you say your Mac works, is the Mac wireless? What happens when you turn off the Mac's AirPort card and hook the Mac up to the Ethernet cable? (i.e. Any chance you just have a bad Ethernet cable, or it's not plugged into the base station all the way?)