Airport shows connected to a network, the computer has an IP address, but has no connectivity.
Open 'System Preferences’.
-If the padlock icon (lower left of window) is locked, click it to enter your password.
Click ‘Location: ‘ and select ‘Edit Locations’.
Click the little plus sign to add a new location to the list.
-name the location (for example: Default).
Insure the new location is selected (currently in use).
Optional: You can further set the connection priority by clicking the gear icon, and setting the service order.
The issue should be resolved.