How do airports handle traffic during emergencies?
The facility under duress will declare "ATC Zero" and all traffic will be handed off to other controllers while that facility is offline.  

In the case of a control tower, this means the airport either acts as a non-towered field or as a closed field (depending on the nature of the emergency).  If it is a TRACON, then the ARTCC will assume control of the TRACONs airpsace.  If it is an ARTCC, that might present bigger problems as continuity of radio communications may be impacted and I'm not sure exactly how this would be handled but assume it would be re-routes around the ARTCC.

I've been inbound to IAH when their tower decided to close (I forget why), and the response was to put everyone inbound to IAH into holds and handle diverts when fuel became an issue.