I like to sing this song with young children as a parachute game. They seem to "fall down" more carefully when they are holding the edge.
Source: Feierabend, John. The Book of Beginning Circle Games. p. 65
Wikipedia cites reasons for discounting
the traditional connection of the song with the black plague. I like the idea that the "fall down" portion was done as a curtsy. This could be a good song for older children to practice a bow and curtsy.