This is a good question. What you are referring to is the use of the relative pronoun. The two most important points to keep in mind is how the relative pronoun is working in the sentence and how the relative pronoun is working within the relative clause itself. The case of the relative pronoun is determined by the relative clause itself and not the overall sentence.
In this sentence, the correct answer is "who," because the who represents the concert master who does something. In other words, the concert master is the subject of the relative clause, even if the concert master is not the subject of the sentence.In light of this, the sentence should read:
"Playing with the Boston orchestra when she was young, she broke a string; she then walked over to the concert master, who handed her his violin."