Repost of unanswered question
A number of authors have written about the reception of Othello in 18th- and 19th-century America. Tilden G. Edelstein, in a 1982 article reproduced in this book, argued that in the 18th century audiences tended to view it as a play about morality and social class rather than about interracial relationships – although some viewers, such as Abigail Adams, were disgusted by the sight of a white woman with a black lover.
By the 19th century, though, the racial aspect of the play became increasingly important to audiences. One effect of this was that the makeup of the actor who played Othello – who was, before Paul Robeson took on the role in 1930, nearly always white – became gradually lighter and lighter, to the point that it was hardly discernible. (John Wilkes Booth’s brother Edwin played him as an Arab, wearing a moustache and Oriental robes.) Had it been darker, it was assumed, audiences would have reacted angrily.
The racial aspect also colored the perception of Desdemona. John Quincy Adams was as repulsed by the relationship between Othello and Desdemona as his mother, but he added the Desdemona must have been a “lady of easy virtue” and a trollop to devote herself to a black man.
On the other hand, the play was often staged in such a way that reinforced the notion that it was merely a fiction - the actors who played the roles of Iago and Othello would periodically exchange their roles, emphasizing that their race was merely a pretence.
See Tilden G. Edelstein, "Othello in America: The Drama of Racial Intermarriage"; but also Andrew Ian Carlson's 2011 PhD thesis, "How Othello Became Black: A Cultural History of Shakespeare's Moor in New York"; Sujata Iyengar, "White Faces, Blackface: The Production of 'Race' in Othello," in Othello:New Critical Essays, ed. by Philip C. Kolin (Routledge, 2002): 103-32
The thing about opera characters like Cio Cio San and Othello is that they've almost exclusively been played by white performers. Having an Othello in blackface was very common, well after the idea of blackface became offensive. In fact, it was only 2 years ago that the Met made headlines in deciding NOT to put blackface on their white actor playing Othello:
Same with Cio Cio San - It would be rare for an Asian singer to be cast at all, but some like Tamaki Miura could ONLY get cast as Butterfly. In the last 10-20 years, that trend is reversing, and you see a lot of companies who try to cast Asian singers in the principal Asian roles (Turandot, Liu, Butterfly, Goro, etc.). But you'd still, by and large, see nothing but white faces in every opera house across North America and Europe for the entirety of the 50s and 60s.
The bigger question is about Black singers in the US during that time. Shirley Verrett, one of the foremost singers of her time encountered significant racism in being cast at all, particularly in roles that were deemed to be for white singers (insert the irony of being totally fine with a white dude in blackface playing Othello, but being offended by a black Carmen). She wrote a biography about being dumped from performances because orchestras refused to play with her, about having to use "whiteface" makeup because no one how to use lighting for darker skin tones, and all sorts of racist crap that you expect in the Jim Crow era.
So, to kind of answer your question, audiences seemed to have no problem with interracial relationships on stage as long as the actors themselves were white, but black opera singers faced the same struggles actually being seen and cast, and treated well as black performers in other genres.