Even when the threats are anonymous, it would still be illegal to respond to random people as if they were the ones issuing the threats. Let me illustrate that point with another scenario. Famous people often get death threats. Outspoken politicians are especially targeted. (I imagine that John Boehner and Nancy Pelosi frequently get anonymous death threats.) These people may have a legitimate fear for their lives, and they might take certain precautions to protect themselves. But if Boehner or Pelosi were to take out a gun and shoot some random person, just because they were afraid, they would rightly be prosecuted.
In the case of the lady with the shooting range, there are a number of options open to her to respond to her fear. If she really feels afraid running a gun range with Muslims around, she could get into a different line of work, like selling flowers. The act of publicizing a restriction on Muslims is a particularly ineffective one. If she fears for her life because of her debates with Muslims, denying them the use of her gun range is not going to offer her any significant protection. This is like the gun control debate. Criminals will always find ways to get guns, while the law-abiding citizens suffer. In this case the random Muslims who just want to shoot for recreation are the law-abiding citizens who are penalized because of the ones who made threats. But those who made the threats are the criminals, and they are not going to be stopped just because she closes her business to Muslims. In fact that action is more likely to anger borderline Muslims and inspire them to get their own gun to do the job.