Internet is a permanently open window to the world right in your home. Children can't tell how far they can go and how danger peeks at every corner.
Internet can be very helpful in a child's development. On one hand it's a bottomless well of information, on the other, it doesn't disregard the need for research.
Nevertheless, an uncontrolled use of Internet might turn out to be extremely dangerous. So your child takes the best out of the Internet experience, follow some ground rules:
If you still can't get peace of mind you can monitor all of your child's activity on the Internet. Nowadays there's free online software custom made for that effect.