Basically, the government's doing with aliens what they did with 9/11 in that one episode of South Park. During the Cold War the government was testing a lot of aerospace technology that they wanted to keep classified. One of the things they did to make it look secret was give a few winks and nods that they were covering up actual aliens. This probably would have never fooled the Soviets and was probably mostly for the benefit of keeping it out of the mind of most Americans, but it wouldn't require a great deal of resources. They probably figured that they just had to drop the idea, and it would spread so fast in the public consciousness that most people who would care about government spy planes would forget about them and focus on aliens.
After the Roswell Incident, I do remember reading about the military saying that it was a flying saucer the day after it happened, and then they quickly retracted the statement. It turns out that it was a type of spy balloon, and without any real evidence, I think that the first statement was made just because of how ridiculous the original statement would have been without hard evidence. Later in the 1950's, the Air Force created Project Blue Book, a research group devoted to investigating and debunking UFO sightings. The group was pretty heavily criticized for shoddy research. This can largely be chalked up to the fact that the Air Force didn't really give much of a crap about UFO's and barely funded them, but it did provide the side benefit of fueling conspiracy theories. Now, obviously I don't really have any hard evidence for this, and conspiracy theorists don't seem like a huge threat to the government now, but our government has certainly funded resources for more innane things and gone after non-threatening groups. Many military officials might have felt that public attention was inevitably going to focus on any new technology that they tried to build, and that this would be a cheap, easy way of diverting public attention.
Factsheets : Unidentified Flying Objects and Air Force Project Blue Book
Project Blue Book - Wikipedia, the free encyclopedia
Roswell UFO incident - Wikipedia, the free encyclopedia