Working at MSY in an operational capacity for almost 13 years, I've experienced a number of incidents in which some have, and some have not, caused the airport to shut down.
The first was a Continental Express Jetstream that crashed after takeoff, and wound up about 2,000' off the departure end of our rwy 19. At the time of the incident, the airport was totally closed down for a few minutes, in order to give the ARFF vehicles free clearance to reach the crash site. The airport's east/west runway was opened after a short while, while 1/19 was reopened....I think it was after the NTSB got here either late that night or early the next morning.
A couple of years ago, a Champion Air 727 ran off rwy 28 in a driving rainstorm while rolling out on landing. That runway was closed at that time, and up until the 727 was removed (it took 2 days), while 1/19 stayed open.
I would think that the medium-size and smaller airports would probably at least curtail operations, until such time as the determination had been made that it could reopen. Airplanes that were cicling obviously would understand the need to go to their alternate, and wait for reopening.
As for the larger airports, or more spread out airports, DFW, MCO, DIA, etc, it would most likely be a judgement call at the time, with ,of course, the primary determination being safety of the aircraft involved.
The other thing to consider when deciding whether or not to close the airport would be the availability of ARFF equipment to provide service in other areas of the airport should they be needed.
Tom in NO (at MSY)
"The criminal ineptitude makes you furious"-Bruce Springsteen, after seeing firsthand the damage from Hurricane Katrina