Although airliners officially keep very quiet about UFO sightings made by their employees, those same employees are often quite vocal. Their employers would wish they kept quiet for a variety of reasons. With major studies reveal the drunkenness of airline pilots recently released, they do not the public thinking they hire drunk pilots or overwork their pilots.
On January 6, 1995, a British Airways pilot and co-pilot claimed to have seen a UFO over England. It looked like bright lights that sped by at remarkable speeds unknown to modern aircraft. It later turned out to be an experimental RAF craft nicknamed "The Silent Vulcan", which does travel at mind-boggling speeds. The pilots report that although British Airways allowed for them to keep their jobs, they were mercilessly teased by all co-workers.
Although the public has mostly lost confidence with most major airlines for issues, pilots seeing UFOs might just have passengers racing for trains and boats. A lot of the times pilots and co-pilots claim to see UFOs, none of their passengers seem to have noticed anything. That can make even those who believe in UFOs to be skeptical of the pilot's physical state.
Airline pilots have a very stressful job with long shifts. It has only been recently that major American airlines have tried to limit a pilot's shift to twelve hours or less. Before that, they could be working days straight. Fatigue brings perhaps more believable hallucinations than does inebriation. Falling asleep at the controls of 747 is more of a problem than you might think.
Even the National Transportation Safety Board is so worried about pilot fatigue that they have tried to stress to airlines about the dangers since 2001. One of the signs of someone so tired that they should not be handling a pillow, let alone a Boeing, is that they will see flashing or moving lights. This is also a signal that a migraine is on the way, triggered by stress, lack of sleep and possibly caffeine withdrawal.
Airlines would probably love to meet extraterrestrials (perhaps a new target audience?). However, it is not an airline's job to determine whether Earth has been visited by aliens. It is their job to get customers from Point A to Point B as quickly and as safely as they can (and, preferably, make a profit in the process). Since the average airline customer does not believe in UFOs, then the airlines will keep any supposed UFO sightings by their employees quiet.