Kipling's specific reason for writing The White Man's Burden was basically intended as, in Albert Herring's phrasing, 'an exhortation to the US to carry out its imperialist "duty" in the Philippines, which it did, with some enthusiasm and casualties.'
The New York World printed this response when the U.S. was trying to figure out how to treat those territories, such as the Philippines, which it had won from Spain during the Spanish-American War:
We've taken up the white man's burden
Of ebony and brown;
Now will you kindly tell us, Rudyard,
How we may put it down?
This stanza was published without a title, and apparently without a particular author listed. I got it from a page which asked that it be cited as:
New York World. "We've Taken Up the White Man's Burden." New York World, rpt. The Public 2 (July 15, 1899). http://www.boondocksnet.com/kipling/nyworld.html In Jim Zwick, ed., Anti-Imperialism in the United States, 1898-1935. http://www.boondocksnet.com/ail98-35.html (May 14, 2001).