We have asked how a government could cover up an alien crash before, but the real question is why they would want to. Given that in most science fiction series, that the aliens either outright attack or humans join the aliens, there is never really a good explanation for why the government would want to cover up alien existence. Given that in our own world the government claims they are searching for alien life, why would the American government want to cover up the existence of aliens?
This question is pretty close to opinion-based idea generation, two bad things at once. I'll try to give a non-opinion-based answer.
Governments have several reasons for keeping secrets.
- Revealing the secret would disclose means of intelligence gathering. For example, I'm pretty sure that in the real world the NSA could shut down many of the ransomware encryption trojan criminals, but making that stick in court would force the NSA to reveal how they work. So they don't do it.
- Revealing the secret would disclose targets of intelligence gathering. The initial indication of the secret might come from an operation against a neutral or friendly government. Follow-up investigations would be classified at least as high as the initial clues.
- Revealing the secret would undermine official or semi-official ideology. Perhaps the aliens are Communist (or not Communist). Or the aliens are Christian (or not Christian). There is a nice quote from a character in the Laundry novels by Charles Stross how the US is at a fundamental disadvantage against eldritch horrors because of the widespread religious beliefs of their population.
- The government did something evil in their inital reactions to the aliens and now they want to sweep that under the carpet. Perhaps a SAM downed an unarmed flying saucer full of tourists. Or they kidnapped and vivisected a member of the first contact delegation.
- The aliens demand that the secret is kept, for reasons of their own, and their hosts comply.
Either way, once the existence of aliens is kept secret and the first official denials are given, changing this policy would damage political leaders.
One possibility that didn't come up yet: The government is aliens. The aliens don't want to reveal that they are already ruling us, as that will likely lead to much opposition by the humans, if not revolution. And the best way to make sure nobody suspect that the government is composed of aliens is to make sure that nobody believes that aliens even exist, or at least not on this planet.
They could want to cover it up because:
- Concerns over mass panic of their citizens
- To gain a tactical advantage against foreign powers
- Knowledge is power; if I know something you don't I'm at an advantage
- Technology can be researched/reverse engineered that places one nation at the advantage over another.
- To gain a tactical advantage against its own citizens (see above)
- To keep the status quo: Maybe the powers-that-be just don't like what such a discovery would imply to the social change.
Also if the aliens were hostile, it could be used to keep information from the rest of the aliens. The scout ship just 'disappears' and the other aliens may be uncertain what happened to it (destroyed on entry? Missed Earth entirely?).
Kay from Men In Black put it best:
A person is smart. People are dumb, dangerous, panicky animals and you know it.
The best reason to want to cover something up is that people might do something stupid and damaging with that information if it got out. A wise individual knows there is a time and place for everything, and it may not yet be the time for such information.
There are probably as many reasons as there are pairs of earthly governments and alien cultures. A few prominent candidates:
1) Aliens are a source of advanced technology.
All states and state/corporation complexes are dependent to various degrees on control of the most advanced technology available in their geographic area of control, not just weapons but also surveillance/communications, computing, agriculture... Even entertainment. America will be in a bad way the day the world doesn't want to see its movies anymore. Because states depend on controlling advanced technology, and aliens visiting Earth are an obvious source of advanced technology, keeping those who would seek advanced technology to resist or assimilate the state ignorant of the existence of aliens is the first sensible and maybe only possible effective step in keeping alien technology from falling into the hands of another Earthly faction.
2) Aliens, if they are visiting here now, probably have the know-how to dominate us completely, and if they did, the population may not take it well.
Even if they couldn't actually dominate a planet light years from their home, they'd have a trivially easy time making our leaders think they could. It's possible that they would agree to pay great tribute to stave off a full-scale invasion (or even extermination.) But how do you think people would take it if we learned that there is no "1%", at least not in the human demographic: they really are Reptilians or whatever, and they're growing fat on the wealth of our world while they cook us to a crisp! Probably not incredibly well. Maybe badly enough to disrupt planetary production and trigger an invasion, certainly badly enough to threaten the job security of the leaders who agreed to the deal and their political descendants, and that's motive enough for anyone to cover it up.
3) The aliens said so.
If the aliens said "don't reveal our existence to the general population", it is unlikely that they would be ignored, especially when they've cowed us with a demonstration of their EverZapp Brand Intergalactic Space Laser.
The usual rationale is that the government has made an unholy deal with the aliens. More or less like if the US government behaved towards the aliens like third-world governments behave towards the US. Granting unlawful advantages to the aliens, perhaps even sinister ones (where do you think embryos' organs harvested at Planned Parenthood go?), for instance, in exchange for some kind of bribery, or to circumvent blackmail.
It is evidently hogwash, of course, and it isn't even a reasonable plot for fiction. But I fear it is the most common "argument" regarding this issue.