All right, I can't hold it anymore.
First: Without debating the most fundamental point of this topic, let's say bigfoot exist.
Second: Let's assume that the government cannot be so incompetent that it is oblivious to bigfoot.
Third: Let's assume that the government has decided that there are downsides to public awareness of bigfoot.
Fourth: Most people don't believe in bigfoot. Most of those who don't believe in bigfoot don't want to believe in bigfoot.
Fifth: Bigfoot hide themselves, the government doesn't have to hide them. All the government has to do is preserve the status quo.
Sixth: To preserve the status quo requires essentially a passive approach with minimal intervention as needed.
Seventh: The easiest way for the government to achieve this is as follows:
a. Establish internal agency policies and attitudes toward inquiries about bigfoot: If anyone is crazy enough to ask, tell them, "We have no proof that bigfoot exists and science does not recognize its existence." Don't waste any more of your valuable time on the matter.
b. If any of your own people come to you with stories, sightings, etc., simply treat them the same way society treats a civilian who shares his experience with the public. This encourages people to keep their mouths shut for the same reason society encourages people to keep their mouths shut. If someone gets a little insistent, employee peer pressure will eventually take care of it.
Eight: So what happens when bigfoot make themselves known to government employees under one of the three following circumstances.
a. A government employee observes bigfoot activity: Per seven, above, he's encouraged not to waste time on it and to keep his mouth shut, because his peers will be unlikely to believe him anyway due to a dismissive organizational climate. If two or more see one at the same time, it'll be "I won't say anything if you don't." No one wants to be disruptive and jeopardize their job.
b. A bigfoot is disabled, or a body is created, thus it can't walk away: Most of the time, according to various reports, other bigfoot recover the body, so turn your back on it for an hour or so, overnight if necessary, and it's likely to disappear on its own. If it doesn't, then you've got to do something about it, discretely. Use as few people as possible, send your staunchest supporters of agency policy. When someone says, "OMG, they're real!", the staunch point out what a pain in the behind it'd be for all of them if the public knew. The body becomes a big hairy John Doe.
c. A bigfoot does something violent. Most likely it's done and gone, never to be seen again. The violence has to be attributed to something, and officials can simply say that there were victims of a wild animal attack and they are taking steps to prevent further public injury. They can close off the immediate area to hikers and campers and let the thing blow over. No lies told. No conclusive evidence that the public would accept, no foul.
Ninth: It's highly unlikely that the government can control bigfoot. Public awareness would result in public demands that they cannot meet and don't want to have foisted upon them.
Tenth: The official government policy regarding bigfoot need be no more than a simple reflection of public opinion.