Why did Franklin Roosevelt promote Nazi sympathizer/eugenicist Frederick Osborn to high levels of office?

Osborn was very much a part of the "establishment." His eugenics advocacy did nothing to isolate him. Nor did Sanger's views on race and abortion. Osborn was also an associate of Allen Dulles. They worked together to set up early psywar projects such as Radio Free Europe and the Crusade for Freedom. After the JFK assassination, Osborn's son would vouch for Ruth and Michael Paine, associated with Lee Harvey Oswald, to the FBI. Michael Paine's mother, Ruth Forbes Paine, was a close friend of Dulles' wartime mistress Mary Bancroft.

"Eugenics," unfortunately, was a subject that was "accepted" if not popular at the time. But "pro eugenics" was not the same as pro-Nazi, even though there were some overlaps. One Roosevelt ally who was also a believer was a man named Winston Churchill, who was clearly not a "Nazi sympathizer." Most eugenicists advocated "protective" measures toward the "unfit" for the benefit of the rest of society that fell far short of the Nazis' "elimination," and were horrified by the German version.

Apart from his connection with eugenics, Osborn was a member of the establishment. He was a successful Wall Street man (like Joe Kennedy, another suspected Nazi sympathizer). He was an active member of one of the "pet" foundations of the Rockefeller family. Partly "immunizing" him from charges of being "pro-Nazi" (as opposed to pro eugenics) was the fact that some of his closest associates had names like Strauss, and Schiff, and Warburg, and could vouch for him. Finally, he was a member of the Carnegie Corporation; a fellow member was Henry Stimson, Secretary of War, and former Secretary of State, who ultimately nominated Osborn for the Army posts Roosevelt appointed him to.

Osborne, as well as many others at the time, was a believer that Eugenics would lead to a better world for all. Eugenics had become a popular subject well before Hitler twisted it to his goals.

Eugenics was widely accepted in the U.S. academic community.[7] By 1928 there were 376 separate university courses in some of the United States' leading schools, enrolling more than 20,000 students, which included eugenics in the curriculum

It was considered a valid area of research:

The American Breeder's Association was the first eugenic body in the U.S., established in 1906 under the direction of biologist Charles B. Davenport. The ABA was formed specifically to "investigate and report on heredity in the human race, and emphasize the value of superior blood and the menace to society of inferior blood." Membership included Alexander Graham Bell, Stanford president David Starr Jordan and Luther Burbank.

and was also picked up by feminists and those supporting birth control laws:

The National Federation of Women's Clubs, the Woman's Christian Temperance Union, and the National League of Women Voters were among the variety of state and local feminist organization that at some point lobbied for eugenic reforms.

One of the most prominent feminists to champion the eugenic agenda was Margaret Sanger, the leader of the American birth control movement. Margaret Sanger saw birth control as a means to prevent unwanted children from being born into a disadvantaged life, and incorporated the language of eugenics to advance the movement

Roosevelt hired someone who was one of the premier scientists in his field, due to the fact that his scientific knowledge was of import to U.S. policy. Remember eugenics ws being practiced not just by the Germans, but the US as well:

The most significant era of eugenic sterilization was between 1907 and 1963, when over 64,000 individuals were forcibly sterilized under eugenic legislation in the United States.

All above quotes from wiki:Eugenics in the United States

• A little update.

  I looked at the article referenced by the question above, and found this quote:

The German sterilization program is apparently an excellent one,' remarked Frederick Osborn, secretary of the American Eugenics Society, in 1937.

and a little farther down the page, an explanation:

Osborn's enthusiastic endors*m*nt of Nazi eugenic sterilization - which mandated the sterilization of people with disabilities deemed heritable

So this shows that, though the regimes later use of eugenics was horrific, that wasn't what Osborne was referring to. but the attempt to possibly eliminate genetic diseases, not genocide. These are the same applications the US was doing, with most of the sterilizations mentioned above being those in Insane Asylums.