Usually the operator will have a separate and specific briefing for arrivals and departures into a high elevation airport (8,000' 14,500') including modified engine start procedures, an amended MEL, weather parameters, minimum rates of descent (given that the cabin altitude could climb on descent!!), and in some cases donning of oxygen masks for arrival. Depending on the topography the effectiveness of VHF radio communication as well instrument beacons could be compromised.
Dependant on aircraft type the airport elevation is programmed into the presurisation system via the overhead panel/FMC/FMS for high altitude mode before landing at that field. This will modify cabin flow rates and outflow valve operation in line with the airport's elevation and avoiding a cabin altitude warning (could activate above 8000'/8.6psi) and continued pressurisation of the cabin on landing (otherwise its difficult to open the doors for disembarkation/emergency evac).
The legal requirements for basing and operation at such fields are, I imagine, complex and vary from country to country, but some might include no of years of operation of operator, power system reliability of aircraft and maintenance to the equivalent of 120min ETOPS standards. For flight crew: age of commander, experience of the FO, pre-requisite ground & sim training (may include physiology observations and procedures to avoid dehydration) and no alcohol within upto 48 hours may form some of the stipulations for flight crew standards.
I am sure someone, who has more experience in this matter, will correct me if I am awry with some of the above general thoughts (I'm sure there are many more considerations) - hopefully it provides some initial food for thought though...
PP