It's glossed over in the comics, but if it were expanded upon...
I've been looking at this quite a lot lately. One convenient thing is the time period, there were no camera phones or anything like that. When the Rodney King beating happened in 1991, it was an incredibly big deal because it was the first time such a blatant abuse by a large domestic police force had been captured on a camcorder for all the world to see.
In the comics, the first explosions happened in a seedy place in the Bronx and the ships never even decloaked. Philips and his troops had already been onsite in the city monitoring things as they were unfolding and that would be an easy enough cover up. There were also several clandestine organizations active in that period that had been tracking Yautja activity and retrieving their tech for years, if not centuries up to that point.
So the big encounter happened at Mid-town near the Macy's on 6th. Once again all the ships were cloaked until Schaefer blasted one out of the sky, and then later when one decloaked near the ground above the crowd of cops, troops and gang members. The army would have already set up a perimeter. Limiting and containing anyone with direct exposure. So at this point you're dealing with a bunch of hardened criminals, the military, the police force, and the bystanders. Things happened pretty fast and once the rain started pounding not only would a ton of evidence been washed away or contaminated but, the Yautja end up booking it shortly after things escalated to the point where the Choppa's showed up. Everyone within the perimeter would have been rounded up. The Army would have gone full Roswell coverup on the situation. The destruction would be chalked up to terrorist activity involving a stolen Macy's day balloon, or a blown gas main and mass hallucination. The gang members would be discredited and imprisoned or disappeared. The troops involved would all be subject to classification. And the rest of the civilians would all be paid off, threatened, silenced or discredited. The unbelievers would attend to themselves and be convinced it was all a hoax, and the others would be written off as kooks, or paid or threatened into silence.
The biggest crinkle would be the wreckage from the ship that Schaefer blew out of the sky. According to subsequent stories Philips was under the impression that the army was unable to recover any hard evidence, but it's no secret that Philips screwed up the whole operation, and subsequently he was compartmentalized and excluded from all Intel regarding the major incident in New Mexico later that year and the tech and body that was recovered there. As for any recovered wreckage from the destroyed ship in midtown, the Army either did or didn't round it all up. If they did, then Philips was kept out of all knowledge regarding this, and someone more connected was doing an end run around him or keeping him around as a patsy. It's stated in the comics that the president specifically does not hold him in high regard and it already seems like Philips is being managed from higher up. If the army wasn't responsible for confiscating the wreckage and other bits then the question becomes, who was? Surely an organization with plenty of resources. The Borgia syndicate, the Soames family, or even Gideon Suhn Lee are all well connected, likely monitoring the situation better than the Army was at that point, and have all been shown to have acquired extensive collections of Yautja tech and artifacts over the years, decades, and centuries.
