This is another one of those "dated" qualities about the show. You can't avoid it, because it was produced in the 1960's.
Correct, and an accurate reflect of the social and political climate underneath the philosophical ideal the show creator tried to promote.
However, if done TODAY as a TV show, there would more than likely be black captain in command, a female of ambiguous sexual orientation as his 2nd, our former 'enemies' would now be our 'friends', and the ship would contain only 17% White Anglo-types in either positions of limited respect and responsibility, or serve primarily as the series Villains,.... oh wait,. they already made that show too. LOL!!!!