Responding to an anonymous letter is not like boxing with a shadow. A shadow is inanimate and inactive. Writing or receiving a letter, whether or not you sign it or it is signed, is still taking an active role. Unless you have ever interacted with a shadow that writes and presents arguments, this situation is not like boxing with a shadow.
Not responding to anonymous letters is a passive response. Not responding is sulking.
Responding to anonymous letters (while I do fault the original accuser for failing to identify her or himself) is proactive and offers some attempt to create a dialog, to address concerns, to present various viewpoints and perspectives. Responding to anonymous letters allows you to empower yourself, even if the person who wrote the letter has decided not to empower him or herself.
Perhaps this person is remaining anonymous because she or he feels threatened or has been emotional or physically attacked recently. Perhaps this author is afraid that she or he will be personally attacked or harassed for writing this letter. There are some instances in which it is acceptable, and in which I would argue, it is best to remain anonymous. Perhaps you have information about a drug lord who has been killing people, wouldn't it be safer for you to report this anonymously? If you are the of the non-dominant gender, race, sexual orientation (or other signifier) it may be unsafe to raise accusations if the person you feel attacked or threatened by has dominance or power over you.
Do I think it is better to sign your letters? YES. Do I think it is better to respond to an anonymous letter rather than do nothing? YES. Do I think it is important to take a proactive role in contributing to a safe environment that encourages and creates dialogs about concerns? Absolutely.