(written from a Production point of view)
Fu Manchu was a major influence on the facial design of the TOS Klingons, at the suggestion of actor John Colicos. Instructions he gave to the TOS makeup department about how to create his Klingon look included the direction, "We'll borrow some stuff from Fu Manchu, and put a long moustache and eyebrows on me." ("The Sword of Colicos", Star Trek: Deep Space Nine - The Official Poster Magazine, No. 8) The Klingon make-up scheme therefore originally had "a little touch of Fu Man Chu [sic]," in Colicos' words. (Cinefantastique, Vol. 28, No. 4/5, p. 59) Michael Westmore agreed that the moustaches of the TOS Klingons "had the Fu Manchu look to them." ("Michael Westmore's Aliens: Season Two", DS9 Season 2 DVD special features) In the script of TAS: "The Time Trap", Klingon officer Kuri was described as having "a Fu Manchu beard and mustache."
Fu Manchu is also referenced in "Inside the Roddenberry Vault, Part I", a featurette in Star Trek: The Original Series - The Roddenberry Vault; in that documentary, Michael Okuda refers to a particular group of movies – which were discussed in a deleted scene from TOS: "The Corbomite Maneuver", and were said to have featured villainous Orientals – as "old Fu Manchu movies," even though Fu Manchu himself wasn't specifically referenced in the scene.