If you use the built-in modem on your AirPort base station to connect to a dial-up Internet Service Provider (ISP), you must initiate the connection attempt from a wireless client. Clients that are connected to the base station LAN port over Ethernet cannot trigger modem dialing.
After the dial-up connection is established, wired clients will be able to use the connection; they just can't start it when the connection is not already open.
This issue affects all AirPort and AirPort Extreme base station models with a built-in modem.