Ms. Copeland said she had been pleased to see more racially diverse audiences turn out at some of her performances in the past year. “From the day that I met my manager, Gilda Squire, she asked me, ‘What do you want to do?’ ” she recalled. “And I said, besides continue dancing at A.B.T., I want to bring more people to ballet, I want to see more people that look like me on the stage, in the school, and in the audience — on the board.”