BIG +1 herewhat follows is an opinion.
As a teenager, although (mercifully) close to exiting that phase, i disagree.
The internet is a fantastic way of maturing in an environment that doesn't blame or leave permament problems. Restricting access to a computer just creates an urge to 'find out' what exists in the shadier areas of the internet that you wouldn't visit under the watchful eye of a parent. If you've got access to it, you won't feel compelled in the same way.
Things i would suggest instead.
1. Instruct them in the risks of real names and locations. Ideally, impress upon them the importance of going under an alias until they are old enough to make decisions about which parts of themselves they show online.
2. Make them aware of the laws regarding the internet. Discussion of the sentences involved (and how disappointed in them you'd be?) is the best way of discouraging dubiously legal activities.
3. Restricting time spent on a computer is a sure-fire way to make them annoyed about their lack of freedom, having just discovered a world on the internet. However, encouraging them to do something constructive at least part of the time is a good thing as it leads to more positive things for them to do on a computer.
Also, bear in mind how it is growing. Do you really wish to have them naive about the internet in today's world? In tomorrow's world it might be the difference between a job and the employment office (not saying it will be, but it's looking more and more likely).
Just a couple of thoughts that you might like to consider before imposing a restrictive regime.