How is the "burden" influenced by the scientific thought at the time?

At the time that Kipling wrote "The White Man's Burden," the prevailing science—or more accurately, pseudoscience—held that white people were racially superior. It was widely believed by scientists as much as anyone in society that white people were possessed of inherent genetic advantages that made them fit to rule over those deemed inferior.
Such conclusions were used to justify the colonial project, which was presented—in "The White Man's Burden," for example—as a benign, selfless effort on behalf of the white man to bring the benefits of Western civilization to the poor, benighted natives of what is now called the developing world.
Kipling's low assessment of the indigenous peoples' intellectual and moral capacity—"half devil and half child"—is closely related to how the prevailing scientific consensus regarded colonial subjects. Scientists, no less than Kipling himself, believed that the so-called inferior races were congenitally incapable of self-government—that they would need the firm, but benevolent hand of white colonialists to run their affairs for them.
But not just any white colonialists; only the "best ye breed." This implies a belief on Kipling's part in the then popular pseudoscience of eugenics, which held that it was possible to improve the genetic composition of the human race through selective breeding.
http://www.kiplingsociety.co.uk/poems_burden.htm