"amorous," 1900, perhaps connected with goggle, because the earliest reference is in goo-goo eyes. Use in reference to politics is from 1890s and seems to be a shortening of Good Government as the name of a movement to clean up municipal corruption in Boston, New York, etc. It soon was extended to mean "naive political reformer." Goo-goo as imitative of baby-talk is from 1863.
as a noun, a cricket term, 1903, of unknown origin. As an adjective, of eyes, 1901; see goo-goo.
1899, U.S. military slang for "Filipino" during the insurrection there, probably from a native word, or imitative of the babbling sound of a strange language to American ears (compare barbarian). The term goo-goo eyes "soft, seductive eyes" was in vogue c. 1900 and may have contributed to this somehow. Extended over time to "Nicaraguan" (U.S. intervention there early 20c.), "any Pacific Islander" (World War II), "Korean" (1950s), "Vietnamese" and "any Asian" (1960s).
1893, American English slang, from goo + -y (2). The first element perhaps somehow imitative, or shortened from burgoo (1787) "thick porridge."
"a person who actively seeks to correct social ills in an idealistic, but usually impractical or superficial, way," 1650s (as do-good), in "Zootomia, or Observations on the Present Manners of the English: Briefly Anatomizing the Living by the Dead. With An Usefull Detection of the Mountebanks of Both Sexes," written by Richard Whitlock, a medical doctor. Probably used even then with a taint of impractical idealism. The verbal phrase do good "do good deeds" was in Old English.
Modern pejorative use seems to have begun on the socialist left, mocking those who were unwilling to take a hard line. OED has this citation, from The Nation in 1923:
There is nothing the matter with the United States except ... the parlor socialists, up-lifters, and do-goods.
The form do-gooder appears in American English by 1922, presumably because do-good was no longer felt as sufficiently noun-like. A slightly older word for this was goo-goo.