If you take a look at the specifications of the D600, you will see that it is wireless "b" which requires older WEP security settings.
http://www.cnet.com/products/dell-latitude-d600/specs/
Unfortunately, the newer AirPort routers dropped WEP about 3-4 years ago because it is no longer secure, and easily cracked with free tools available over the Internet.
As a test, turn off the wireless security on the AirPort router and see if the Dell can connect that way. If it can.....then you know that the Dell is not compatible with the higher levels of security that the Airports and other newer routers are using.
Not recommending that you do this, but you could try lowering the security on the AirPort down to WPA/WPA2.....but I don't know whether the Dell will connect that way or not. It won't if it is limited to WEP levels of security.
Best to keep the security at recommended levels and try a USB wireless adapter to see if that will work with the Dell.