Most if not all cable providers authenticate their users by checking the hardware address of the network card attached to the modem. If you were originally set up with just the computer connected to the modem, it sees the AirPort device as a different computer, and refuses to give it a connection.
Turn off the modem and the ABS. Connect the ABS to the modem. Turn the modem on and wait about two minutes (for it to come on and initialize). Turn on/plug in the power for the ABS. If the modem is properly configured (by the ISP), it will read the ABS's hardware address, add it to its list of acceptable hardware, and connect. If it doesn't, you'll probably have to contact your ISP's tech support and ask them to manually add the address (which is on the label on the bottom of the ABS).