It attracted some criticism over the size of Cooper’s nose, which some social media users said played up to offensive Jewish stereotypes.
But Bernstein’s family said they were “perfectly fine” with Cooper using make-up to “amplify” his appearance.
There has also been criticism that a Jewish actor was not cast to play the West Side Story composer.
“It breaks our hearts to see any misrepresentations or misunderstandings of [Cooper’s] efforts,” wrote Jamie, Alexander and Nina Bernstein in a statement posted online.