I served as an Altar Boy for many years, before Communion in the hand was permitted. I witnessed several dropped Body of Christ incidents. All but one I caught with my paten ... the priest simply picked the Sacred Host up and gave it to the communicant.
Then there was "The Disaster": Father dropped the Ciborium! That, I couldn't catch ... He picked up all the Hosts, told me not to move, went to the Altar and picked up a purificator, with which he covered the spot on the floor. We went on, being careful where we stepped. I suppose after Mass, he cleaned the floor ... I had to go to class.
In my altar boy days I had one dropped Host. I caught It (deftly, I thought), and Father retrieved It and gave It to the communicant. That was in the days just after VII, when Catholics still received at the rail, on the tongue.
Much more recently, when I was a communicant (at the same -- Jesuit! -- church, as it happened) presenting myself for Communion (standing this time), the priest fumbled his placement of the Host on my tongue, and It fell to the floor. I looked at him. "Well, pick it up", he said. So I did, and consumed It. I very much regret this now. When I related the incident to a friend who's an Episcopalian priest, he scolded me: "You screwed up. You should have answered him, 'NO. YOU pick It up.' That's his job." He was right, and I pass on his advice to this thread at large.