downbeat n.

"in a descending direction, from a higher to a lower place, degree, or condition," late Old English shortened form of Old English ofdune "downwards," originally of dune "off from (the) hill," from dune "from the hill," dative of dun "hill" (see down (n.2)). The "hill" word is general in Germanic, but this sense development is peculiar to English. As a preposition, "in a descending direction upon or along,"  from late 14c.

To be down on "express disapproval of" is by 1851. Down home is from 1828 as "in one's home region," as an adjective phrase meaning "unpretentious" by 1931, American English. Down the hatch as a toast is from 1931. Down to the wire is 1901, from horse-racing.

Down Under "Australia and New Zealand" attested from 1886; Down East "Maine" is from 1825; Down South "in the Southern states of the U.S." is attested by 1834. Down the road "in the future" is by 1964, U.S. colloquial. Down-to-earth "everyday, ordinary, realistic" is by 1932.

"with a positive mood," 1947, apparently from on the upbeat "improving, getting better," attested from 1934 and a favorite of Billboard magazine headline-writers in the early 1940s, from the musical noun upbeat (1869), referring to the beat of a bar at which the conductor's baton is in a raised position; from up (adv.) + beat (n.). The "optimistic" sense apparently for no other reason than that it sounds like a happy word (the musical upbeat is no more inherently "positive" than any other beat).