Assuming that you mean the 20th century in general, rather than the period from 1900-1910 specifically, I have a previous answer that talks about a similar topic matter (except specifically in the pre-Civil Rights era before the 1960s) which I've lightly edited here:

Black musicians did have plenty of success in the pre-Civil Rights era but there were significant caveats to it. Firstly, I think it's fair to say that their music looks significantly more popular in retrospect. Obviously, forms of music like 'rock'n'roll or jazz or the blues had significant input from black artists - but very often, the most commercially successful artists in those veins were white. However, in terms of the history of that music, the most commercially successful artists are not as important as the most influential artists, and the most influential artists tend to be more likely to be black. For example, some of the most successful big band jazz bands were those of Benny Goodman and Glenn Miller (both white bandleaders), but a history of big band jazz would probably focus much more on Duke Ellington or Fletcher Henderson (both black bandleaders). These musicians were not as commercially successful - and their skin colour didn't help there - but they were more influential in one way or another. Henderson and his orchestra is often seen as a particularly important figure in jazz's transition from Dixieland to big band jazz, but he eventually ended up selling his arrangements to Benny Goodman and writing further arrangements for him. Duke Ellington, on the other hand, is known for the originality and wit of his arrangements, and his ability to take the form to artistic new heights (along with the joyousness of the interplay between some of the musicians he had in his band over the years).

Also, with black musicians, it was very common for white acts to have bigger hits with the same song originally done by black artists - quite often, the white act would record a version of the song while the black artists were still in the charts, and effectively usurp their chart position. The canonical example of this is Pat Boone covering 'Tutti Frutti', originally by Little Richard, in 1956-1957. According to Joel Whitburn's book Billboard Top 10 Singles Charts 1955-2000, Little Richard's version didn't reach the Top 10, whereas Pat Boone's version spent 10 weeks there. Similarly, on February 19th, 1955, the Penguins' doo wop tune 'Earth Angel' reached #8 in the charts; in the chart at #8 the next week was a version of the same song by a white act called The Crew Cuts. So there was clearly a group of people buying singles who were more comfortable buying The Crew Cuts than The Penguins, and some of that has to be due to racism.

Of course, to divide American history into 'racist days (pre-1960)' and presumably 'non-racist days (post-1960)' is a definite oversimplification; the boundary in the 1960s is a change in legislation (the Civil Rights act) rather than necessarily the amount of racism (Jack Hamilton's recent book Just Around Midnight explains very well the various racial beliefs that are often implicit in 1960s rock). And racist beliefs are often more complicated than "I hate those people!"; there's usually a particular set of stereotypes about the capabilities and weaknesses of the other race.

In the case of black Americans, there are stereotypes about the innate musicality of black people mixed in with other less pleasant stereotypes (as a result, white people often tend to be most fascinated by black music that seems like it involved little training and a lot of talent, rather than more carefully-constructed, artfully-composed skillful black music). As a result, white people have for a very long time been fascinated by black music; witness the minstrel shows that for a long time were more or less the dominant popular music form in America, which were based on white people dressing up and acting like black people. Their popularity in the late 19th century and beyond clearly doesn't mean that those white people weren't racist; blackface is a taboo today because minstrelsy is very widely seen as racist beyond the pale these days. Which is to say you can be fascinated by the music of people who you hold a low opinion of in general, and possibly even buy their records sometimes; people can believe that someone else is baiscally subhuman, but basically appreciate their talent in a specific arena (music or sports for example). You don't have to care about someone's humanity to dance to records of their music.

Similarly, there were versions of being black that many white people found acceptable, specifically those associated with the term 'Uncle Tom', derived from the title character in Harriet Beecher Stowe's anti-slavery book Uncle Tom's Cabin. Uncle Tom (in African American popular culture, more so than in the original book) came to be seen as a passive, polite character who was content with his lot as a slave, who knew his place (i.e., who didn't expect to be treated the same as whites). Louis Armstrong, for example, was often criticised by people in the black community as an Uncle Tom - his public persona was as an always happy gentle soul who was non-threatening. Armstrong, of course, saw his fair share of racism and bristled against it - as you would expect - but he saw himself as breaking down boundaries, by being let into places that black people had not previously been and getting some people to see his humanity. In contrast, critics of Armstrong in the black community saw the Uncle Tom persona as severely limiting; they didn't want to subsume their personalities to the very limited public personae that were allowed in public for the likes of Armstrong, and instead wanted to advocate for more authentic modes of being black in public.