OK, thanks for the info.
If the AirPort Extreme is setup to "extend" a wireless network, then all of the Ethernet ports on the device behave as LAN ports and all of them are enabled.
We asked what operating system you were using on your Mac, but will guess that it is a recent version. So, please check the following:
Open AirPort Utility on the Mac.......(Finder > Applications > Utilities > AirPort Utility)
Click on the picture of the AirPort Extreme
Look in the smaller window that appears to see if there is a listing for Status. If there is, what is the message there?
Otherwise, click Edit in the smaller window
Click the Wireless tab at the top of the window
Check the setting for Network Mode. It should be set to "Extend a wireless network"
If that is the setting, connect a short, spare Ethernet cable.....an length will do.....from one of the LAN <--> ports to the Ethernet port on a computer. Turn off the wireless on the computer and check to see if you can get a good Internet connection that way. If you can, then you know that the Ethernet port on the AirPort Extreme is working correctly, and you have a problem somewhere else.
If you cannot get an Internet connection when you run the test just above, try another Ethernet cable or two just to be sure that you do not have a bad cable......which happens all the time.
If the Ethernet connection is still not working, power off the "main" and "remote" AirPort Extremes, wait a minute, power up the "main" Extreme and let it run 2-3 minutes, then power up the "extending" Extreme and check it again.
If things still do not work, then you likely have an AirPort Extreme with a defective Ethernet switch. Check the ports to see if any of the will work. If not, you will need to exchange the AirPort Extreme.