1876 (n.), in reference to downward stroke of a conductor's baton; 1952 (adj.) in figurative sense of "pessimistic," but that is probably via associations of the word down (adv.), because the beat itself is no more pessimistic than the upbeat is optimistic.
1. Its headlines were suitably downbeat and don't scream out.
2. They found him in gloomy, downbeat mood.
3. The overall mood of the meeting was downbeat.
4. This has made him the ideal investor a decidedly downbeat era.
5. Following the goalless draw against Liverpool, Jos é Mourinho's response was understandably downbeat.