Every now and again on the Internet (or at least the slice of the Internet I keep up with), the identity question comes up: What name do you use online? Just a few days ago, I saw it pop up again. In my experience, the person bringing up the topic usually falls on the side of "I use my real name, and so should you." Also, for what it's worth, that person tends to be male.
I think blanket anonymity is problematic. To carry on a discussion it's simply convenient to have some identifier, so you don't have multiple people listed as "guest" or "anonymous" getting confused for each other while expressing differing opinions, for example. But beyond that, anonymity gives people the freedom to say exactly what they think. That can be a real benefit (we use anonymous ballots in the U.S.), but it can also cause lots of problems (see: trolls).
More common are pseudonyms. I'm a big fan of Dr. Isis. She's built a reputation on an unofficial name. Under that name, she can say what she pleases without as much risk of colleagues, co-workers and students taking issue with her opinions. It may appear to be a flavor of anonymity at first, but I think that element of reputation makes a big difference. If Isis were hateful, she'd have a completely different reputation, and her readership would be very different as a result. (Thankfully, she's pretty awesome.)
Then there are those who insist on "real" names. The common argument is that anonymity/pseudonymity breeds trolls because there are fewer consequences for the negative things they say.1 The great advantage to using your real name mirrors an aspect of anonymity: the anonymous can shed responsibility for disreputable behavior; those using their real names can take credit for their good works.
There's an aspect to the use of real names that bothers me, though, and that is not brought up as often as I'd like. If I go by my own name, I am revealing more than what the world calls me. A name can reveal (or hint at) gender, age, ethnicity, religion, and status. John Smith, Bertha van der Beek, Eli Goldberg, Maria Gomez, Margaret Fortescue-Asquith, Jyoti Singh, would all leave different impressions. Sadly, noticing those differences doesn't always work for the better, either. I think anonymity (and pseudonymity) can be a great equalizer. Your words stand on their own without prejudice from your name or title. With pseudonyms, you get to choose what aspects of yourself you reveal.
Before I started this site, I considered whether to attach my name to it. I worried about unwanted attention, as many women do. I worried that any one of my opinions might be unwelcome to potential colleagues or employers. I worried that I might reveal too much of myself. I decided to use my name here, though, because I wanted to be able to say "here is something I have done; here are words I have written." I decided that others' bad experiences were not going to scare me away from writing. I decided that I want to work with and for people who like me, and that hiding this part of myself from them might not be a great way to form strong relationships. I also recognized that there aren't many things I'd say that I wouldn't want tied to my name. The balance made sense to me. I know, though, that the balance tips the other direction for lots of other people. I just wish more real-namers recognized that.
1: This is aside from the fact that people can be plenty rude and obnoxious without the cloak of anonymity.