How high do they cut the grass on airports?
It depends on the type of bird and mammal species in native to the area and also the type of grass used in turfing; some of them may not survive if mowed very closely. FAA's Wildlife Hazard Management Manual concedes:


  Species-specific responses may be expected.  ...  Airport managers 
  need to work with airport wildlife biologists to determine what species of concern in regards to aircraft safety are in their area and what the habitat needs are for those species.   


For most airports, FAA recommends a height of 5-8" of grass.


  Considering bird, mammal and vegetation limitations, mowing at least monthly at a target of 5 - 8 inches may work in many airport environments as 
  part of a wildlife hazard management program. ... Mowing at 5 - 8 inches should also reduce 
  small mammal abundance.