Some people would say that computers are so dangerous that children should not be able to use them. While this may be true in some cases, computers are simply a fact of life in American society.
Computer skills will be required for most gainful employment and will certainly be necessary to perform the functions of desirable careers that we wish for our children. As such, we need to make sure that computers are not hidden from the children. We need to provide access and instruction that is positive and responsible at the same time. To repeat what is often both said and ignored, "children learn by example" and we are their teachers.
There are a few things to consider when getting your children started with computers and if desired, providing Internet access.
Location & Internet Access:
In many cases, the family computer is already located in a centralized portion of the living area. If this is the case, you're halfway there. If not, consider that your work is lessened if the computer is place in a centralized living area where the screen is viewable to others in the room (ie:parents/guardians) w/o anybody having to sneak around or lean over shoulders.
This allows the user to understand that their activity is not secret while providing a sense of limited privacy as others are not constantly leaning in over their shoulder.
If the computer has unfiltered Internet access, this is also necessary should any unapproved content find it's way to the screen. There are many ways to filter access, but in all cases, it's best to have the screen viewable without taking invasive actions that invade the immediate personal space of the user.
Internet access can be filtered in a variety of ways. There is software that can be installed in the computer. There are hardware/software solutions (such as ipcop) that can be installed at the router. In extreme cases, there are spyware programs that secretly log the activity and send it to the parent.
Children with laptops present issues with relation to computer placement. You obviously can't see the screen if your child is sitting in the closet under the stairs or hiding under their covers at 3am. Laptops are also easy to hack past the filtering software since they can easily be handed over to a friend with a little bit of knowledge on the subject.
In summary, pay attention to the computing environment, make sure that the location of the computer lends it's self to access and accountability at the same time. Try not to use spyware as it diminishes trust and leads to advanced bypassing skills at an early age. If you want to be honest about stopping bad content, install something like ipcop and let the whole family live by the rules.
HW/SW solutions like ipcop can also be configured to filter traffic to specific computers at specific times. One can limit access for certain computers from bedtime till walk up and prevent midnight chat sessions with pedophiles pretending to be classmates.