This is a really big discussion.
It goes far beyond the more mundane matters (what matters to us, honestly) like anonymous trolls annoying us and such. This is kind of obvious, surely, but I think it's important to point it out anyway.
Ok, so... The important question: what do I think about anonymity on the internet?
A: I believe it is fundamentally necessary for the internet to work as it was intended to work. It can be misused, true, but cars can be misused too. You can run over people or crash them on other people's cars. In our end, it's all a matter of decency - like Granger points out. But on the greater schemes of things, the benefits of anonymity are far greater than its problems.
Now, a more in depth explanation.
Some here might be aware of the hacker culture. I mean... Not the "omg, I'm invading sites" culture, but the actual hacker culture - the one that defined the term back when the internet was giving its first steps. The one that dialogues directly with the notions of open software and all the cyber- things (cyberactivism, cyberculture, etc...)
Ok, so... I'm not going over all the people and theories and everything behind all that

There's literally a TON of stuff you can read on that - and a lot of interesting people to talk to and that are easily accessible on twitter and other social media. But a couple of names I can mention out of my head are Lev Manovich, Steward Brand, Richard Stallman, Andy Cameron, Axel Bruns, Lucia Santaella, Henry Jenkins... Hell, even Deleuze can come into the discussion with his notion of a society beyond the society of control... Anyway, and, of course, the pioneers behind the internet - "the first hackers" - and things like that, like Licklider, Welden Clark... even Ken Kesey with his crazy "Merry Pranksters" band helped, lol. (Look that up, hilarious story)
Anyway... It's recent history, there are TONS to look up. My point is, though, that the basic foundations of the internet - contributed and observed by many of these people - had the ideals of a free, diverse and non-centralized network for the sharing of knowledge and all that - even though it's partially born as a military network.
But it's to defend such ideals that we have, for instance, universal and open-source protocols (like HTTP) that are the protocols used for connections.
And the two basic principles that "guide" these ideals are:
1. anonymity
2. equity of packets
I'll quickly explain 2 before heading into 1. Packets are the "pieces and bits of information" that we are always sending and receiving. Packets can be identified - that is - governments and surveillance agencies can tell from where packets come from and to where they are going, and what they contain. That's how, for example, a country can "block" certain websites. And not only it's a matter of personal freedom but also a matter of political control - we just need to look at how things like information control are enforced in zones of conflict, how governments forbid people to upload videos asking for help or telling the truth about the situation wherever they are, etc...
So, equity of packets means that every packet should be "equal" or - in other words - impossible to be told from each other and henceforth impossible to be controlled. So information can traverse around freely.
And anonymity, the 1 principle, makes sure that not only information can be exchanged freely but also the people exchanging it are protected. So we can have amazing projects like Wikileaks or some media groups that "take the fight" and start uploading, anonimously, material sent by people in zones of conflict and such.
So this is the discussion, because, obviously, one could argue that these things can also be used by international trafficking, slave trading, child pornography, etc...
And yeah, they do, too. But like I said - cars run over people.
We don't need to talk about prohibiting cars,
we need to talk about educating people.