Letting your kids be on the internet can be scary. You can keep an eye on what they're up to and there are ways to help keep them relatively safe online. However, you might also want to take a page from schools that provide internet access and have kids sign an appropriate-use contract too.
Signing a contract with your kids provides a few benefits. First, it spells out exactly how you expect them to behave because it forces you to explain your position clearly. Second, the act of signing a contract helps drive home the point. And finally, if they do break your rules, there's no argument involved. You can point to the item on the contract that they broke and issue the appropriate punishment (which should also be spelled out in the contract).
Obviously, what you allow and don't allow is entirely up to you as the parent and you might want to give some thought to what's appropriate based on your kids' ages. Maybe they're allowed their own private computer, maybe not. Maybe you give them privacy (at least until they give you reason not to) or maybe not. That's your call.
The folks at The Modern Parent put together this sample contract, which you could just print and use. But we encourage you do to a little research and customise it to your needs. We'd also encouraging spelling out the punishment a little more clearly, though banning them from using the technology they misused seems pretty appropriate and will certainly drive the message home.
An Internet Contract for families: teaching our kids to play safe [The Modern Parent]