It's not only that the Internet let's anyone insult anyone else, it's much more the fact that the Internet is millions upon millions of anyone insulting anyone else.
It's purely a problem of scale.
Libel laws were effective up to the first http connection because before that, the only guy to slander you was a neighbor or someone in your rather immediate surroundings. And you would drag him to court over it because the slandering happened in your surroundings.
Now, your immediate surroundings has been artificially expanded to include countless faceless people you don't know, don't know you and will never meet you, but have no fear of commenting on what they read on Twitter or wherever. And since they have the understanding of a goldfish, their reactions are knee-jerk level at best. Unfortunately, their attention span is longer than that of the goldfish, and some can be quite boneheaded about it.
On top of that, it's a lot easier to be relentless when all it takes is sitting behind a keyboard. The Internet has birthed a whole new generation of stalkers of all kinds, and that is a sad fact.
In all this, the law is now swamped and totally overrun by this potential. Libel laws were designed when one person could be found guilty of slandering another. Today, millions can potentially be found guilty of slandering the same person. How to manage that without locking courts up with slander cases until the end of time, excluding more important things like criminal assault, homicide and kidnapping ?
And you can't really say that people are just going to have to thicken their skin - sadly there have been suicides due to this online behavior. Even though that would be the best solution.
I have no idea what the solution is, but it seems to me that the law is not going to help here. This is a social issue that will just have to find a social solution, not a legal one. And, as much as I don't really like the idea, it seems to me that online anonymity is going to be the casualty of this situation.
After all, the evil slanderers and stalkers do their deeds mostly because they think they cannot be traced - until the cops show up at their door with a warrant. So maybe if they knew beforehand that everything they do and say on the Internet can be immediately pinned to their name and address, maybe then they would think twice before gratuitously insulting and harassing someone.