If the old base station was using the same wireless network name and password as the new Time Capsule, this will confuse the iPhone since it uses a much simpler operating system to scan and connect to wireless networks.
Might that be the case?
My phone connects to other networks fine, so it has to be my base station.
If other wireless devices are connecting to the base station......then it is not your base station. It sends out the same signal to all devices. For some reason, the iPhone is not communicating correctly. I have had similar issues....it was the iPhone that was at fault.
There are plenty of wireless issues with iOS6, especially if you updated from a previous operating system. The wireless issues strike at random times, maybe something else to keep in mind.
If you reset the AirPort Extreme, there is nothing else that we can tell you to do with that device.
If you have not told your iPhone to "Forget This Network" to clear the current...and likely corrupted settings for its wireless connection to your network, that would be the first thing to do. Then log on again to see if the iPhone will connect.
You may have better and more informed answers available in the iPhone support area, so you may wish to post over there. We are not iPhone experts here.
Using iPhone
The experts over in the iPhone support area will also likely suggest this Apple support document on Troubleshooting iOS WiFi Networks and Connections. You can get a jump on things by working your way through this prior to posting in the iPhone support area. Good luck.
http://support.apple.com/kb/TS1398
