When we see the word "both", we should look for parallelism - the two things that "both" refers to must be grammatically parallel. We'll look at A, B, and E:

A. ... created almost two hundred dance pieces both:
--influenced by the works of Armenian theatre director Rouben Mamoulian
and
--Russian-born operatic tenor and stage director Vladimir Rosing

B. ... created almost two hundred dance pieces that were influenced both:
--by the works of Armenian theatre director Rouben Mamoulian
and
--Russian-born operatic tenor and stage director Vladimir Rosing

E. ... created almost two hundred dance pieces influenced both:
--by the works of Armenian theatre director Rouben Mamoulian
and
--Russian-born operatic tenor and stage director Vladimir Rosing

We see that none of these answer choices are parallel. In fact, because the non-underlined portion includes "by the works of" because "Armenian theatre director Rouben Mamoulian" but not before "Russian-born operatic tenor and stage director Vladimir Rosing", there is no way to make this sentence parallel by changing only the underlined portion. So we conclude that we can't use "both" and eliminate A, B, and E.

In C, we see ", who", which indicates we've got a modifier - we should always be able to remove a modifier from our sentence without destroying the grammar of the sentence.

C. American modern dancer and choreographer Martha Graham, who created almost two hundred dance pieces influenced by the works of Armenian theatre director Rouben Mamoulian and Russian-born operatic tenor and stage director Vladimir Rosing; however, the themes she explored were quite different from those of her contemporaries.

C. American modern dancer and choreographer Martha Graham; however, the themes she explored were quite different from those of her contemporaries.

We know that anything on either side of a semicolon must be a complete sentence on its own, but "American modern dancer and choreographer Martha Graham" is not. So we can rule out C for sentence structure reasons, leaving us with the correct answer, D.