I don't expect any child younger than 10 to be allowed to venture onto the internet, so when I use the word child in this post I'm referring to people aged about 10-15.
In truth, the internet is no more dangerous than reality. Yes, a child might end up talking to a paedophile. But they could just as easily be snatched by a paedophile whilst walking in the street, and I very rarely hear of parents banning their children from going outside. What parents should really do is just try and get involved in what their children are doing, instead of outright banning it simply based on paranoia due to what the media tells them about the internet. It honestly just seems like a cop-out. We might as well ban children from eating all forms of candy, in case they get overweight. Of course, if the child is doing things that they shouldn't be on the internet, then a parent has every right to ban it. I say this in regard to the concept that it's just being banned simply because of the idea that it's "omg so dangerous".
I've been on internet forums since I was 11, and I only ever joined harmless communities based around things aimed towards children (such as this site, for example). Not once have I ever had a problem, and that was the extent of my online socialisation. If I were to be banned from such things, I think it would be unreasonable. If someone is posting trivial things in a harmless community, then I really don't see why it's reasonable to ban them from that. In some random rare case that I did end up talking to a scary paedophile, then I wouldn't have been able to meet them without my mum knowing. My mum didn't let me just walk out the door without telling her where I'm going. If parent's aren't keeping an eye on their kids and what they're up to, then that's the fault of the parent, not the internet. The same thing could happen in reality without the internet playing a part at all, and in that situation, the blame would be placed on the parent. If the internet is involved, then suddenly it's all because the internet is a dangerous underworld full of rapists and paedophiles, and parents are helpless to protect their children because the internet has imprisoned the child into a secret world that no parent can access.
The truth is, the internet is just a way to broaden our communication ability, it allows us to make contact with people all over the world. I don't think the fact we can communicate with so many more people online increases the risk, I highly doubt a paedophile from America is going to travel to Europe just so they can abduct a child. Most abductions I have heard about are acted out by people who live in the same area, and if the victim and culprit are in the same area then the abduction could have just as easily taken place without the internet being involved at all. Like I said before, if the parent keeps an eye on what their child is up to then there's minimal risk involved. Even if the child and paedophile did meet online, the meeting would still have to take place in reality, which is something that can easily be controlled by parents.
If anything, the internet is safer because nobody can harm you behind a screen. It's all data, there aren't any portals that open up and suddenly a crazed pedo jumps out the screen. If a child is walking along a street, alone, then anything can happen.
With all that said, I base all of this on the idea that parents are banning the internet solely because they're paranoid of the danger, and can't be bothered to monitor their children so just outright ban it so it's easier for them. Personally, I think there are plenty of other reasons why children would be banned from going on the internet by their parents. Honestly, I think it's likely that I won't be allowing any future children of mine on the internet, or on the computer at all really (unless it's for education purposes). Simply because I regret how I've wasted so much time in front of a screen myself, simply because I have no desire or willpower to even do anything more productive with my time. Wasting time doing trivial things like lurking the internet is all I've known, so it's all I'm comfortable with. I'd much rather encourage my children to do far more productive things with their time, instead of just handing them a computer or video game console.
I may change this view as they get older, because like I have found, I don't make many close friends in reality. So I turned to the internet; I still don't make many close friends online either, but simply because of the vastness of the internet I'm able to find a couple of people that I get along with really well. It's also helped me learn so many things, and I honestly think being on the internet and socialising on the internet has helped me improve myself. But, that only happened in the past year or so, not when I was a child. During my first five years on the internet, I simply wasted my time, trolled people, argued with people, watched silly crap on YouTube and achieved absolutely nothing. I'm not going to let any child of mine do the same thing, and I think the internet can only be truly useful once someone has reached a certain level of maturity and is able to use it properly, instead of just something to play with. I like the fact that I'm on the internet now, even though I could be doing something more productive, I am quite happy with how everything turned out; there are also many worse things that I could be doing, and I don't have many regrets because I'm happy with the results. Although it's near impossible to predict the outcome of something during the time, so even if the same thing could happen with future children of mine, I'm not sure if I'm willing to take that risk. It's all too easy to just fall into a pit of internet addiction and laziness, and end up never gaining anything from being on the internet at all. It totally depends on my child's own personality, maturity, what they want to use the internet for, etc. that will decide when I feel like they'd be okay on the internet.
tl;dr? Paranoia is just silly, although other reasons for banning kids from the internet can be reasonable. The internet is usually only a good thing when people are mature enough to use it properly.
Edit: something I forgot to include, a good alternative in some cases = instead of outright banning, just give children limited time on the internet. That way they won't be wasting loads of time on it, but they're still able to have access. It also depends on how technology changes in the future, the internet we all know of now might have changed entirely in a decade or two.