Traditional telephone harassment laws -- for instance, ones that ban calls (sometimes limited to anonymous calls and sometimes not) that are intended "to annoy, abuse, threaten, or harass any person at the called number" -- are generally thought to be constitutional. As to threatening calls, the First Amendment rationale is clear; they fall within the true threats exception. Likewise, if the law were limited to calls that are annoying for non-content-related reasons (e.g., they're placed at 3 a.m., or they happen twenty times over twenty minutes), it would be justified as a content-neutral restrictions.
But as to "annoy[ing]," "abus[ive]," or "harass[ing]" calls, the rationale is less obvious. The law restricts speech because of its content; the speech doesn't fit within any of the well-established exceptions; and even anonymous speech is usually constitutionally protected. In my view, the best justification for the restriction is that "One-to-one speech that's intended to annoy the one recipient is rarely of very much First Amendment value; people are just rarely persuaded or enlightened by speech that's intended to annoy them. It has some value ..., but to the extent that it's in some measure deterred, the loss to public debate isn't that great -- speakers are still free to speak to others besides the person they're trying to annoy." And the same justification also extends to similar restrictions on harassing e-mail, which have recently sprung up. Yet we should recognize that the First Amendment issue is not entirely clear.
And the question is particularly unclear when the annoying speech is related to public debate, and especially when it's conveyed to public officials. In 1999, the D.C. Circuit set aside a harassment conviction of Ion Popa, who made several racist calls to the U.S. Attorney for the District of Columbia (the chief federal prosecutor in the District); and the court seemed to suggest that speech "intend[ed] in part to communicate a political message" must be exempted from telephone harassment law. (Query, by the way, whether the same would apply to speech intended to communicate a religious message, a message on social issues, and the like.)
In any case, a similar question has just arisen in Washington state:
The Jan. 8 e-mail message, sent by "battleground anonymous" to members of the Battle Ground City Council, was a racist rant about Paul Zandamela, a black man who had been sworn in as a city councilman the previous evening.
"Our city government must be corrupt to have this (derogatory term) as an elected official," read the message in part. The message included four slurs and was signed, "Sincerely, a (derogatory term) hater."
That message and a subsequent note were traced to Christopher Reinhold, the son of Alex Reinhold, Battle Ground's deputy mayor ....
[Mayor Mike] Ciraulo received a second message that called him a "stupid (derogatory term) lover." ...
Reinhold is now being prosecuted for alleged "cyberstalking" based on his e-mail to Zandamela and Ciraulo (not his messages to other council members about Zandamela).
Please keep in mind, by the way, that telephone/e-mail harassment laws are very different from "hostile environment" harassment laws, and thus require different First Amendment analyses.