This advice goes back as far as Usenet, and really is about managing the comments in your weblog. And it's simple advice: don't post any comments to your blog that you find inappropriate.
This means filtering out spam, either by hand or (better) with the help of spam fighting tools; it means taking fake comments from people who are paid to post them and neutering them of their commercial value; and it means taking your angry, abusive critics off your page.
On the spam fighting front, your choice of platform matters a lot. I've been happy with Typepad's service, though it lets a few things go by the admin tools make it possible to unpublish something fairly quickly. People using Wordpress use its default Akismet tool with good success. If your comment tool doesn't aggressively screen for spam, you need a better system. In many cases, Trackback is a source of spam as well; I ended up completely blocking that just to save the hassle.
There's a huge and messy market in fake comments, helped by systems like Amazon's Mechanical Turk that anonymize the payments for them. Every so often the fake commenter actually has something marginally interesting to say, so I tend to let those through but rewrite the target URL of their post to remove the commercial value from it. Life is too short to live with commenter bots.
When on the odd chance that you have an angry, abusive critic aimed at you - the sort of person who would scribble profanities or spew personal attacks - the only reasonable remedy is to remove the text. Ideally, you do that without acknowledging it, or if it's important for context to keep some fragment of it around, a traditional method is to edit the text to make it semi-legible e.g. by removing the vwls or st*rr*ng th* ess*nt**l p*rts **t. Let the critics and nastygrams find their own site, and force their authors out into the open.
One of the reasons that blogs emerged from the Internet primordial soup in the late 1990s was that Usenet had degenerated into spam, fake posts, and abuse; by holding the discussion on a web page you control, you could start to control it. That makes the world a little bit harder to follow when discussions fly around, but in exchange you get to be not just author but (critically) also editor of your own publication.