Is this your aunt's configuration: Phone jack>>DSL Modem>>Airport Base Station??
You're saying that your aunt's mac can surf the net? How is your aunt's mac connected to the airport, via ethernet or wireless?
If it is ethernet then maybe the wireless is password protected? Can you see your wireless icon on the menubar?
If your aunt can surf the net, and she is hooked up via ethernet cable, try connecting via ethernet cable to the Airport. If you can then reach the internet, then it might be the wireless side of the Airport that needs tweaking. You then need to access and configure the Airpot using AirPort Admin Utility.app
Otherwise, have your aunt type 192.168.1.1 on a browser to see if the Verizon DSL modem configuration page shows up, if so, and your aunt can surf the net, you can rule-out that it is a DSL Modem issue.