My kids (ages 7 and 5) have a computer in their room that's hooked up to the Internet. And they're now capable of navigating around the Internet on their own -- on Wikipedia, Google image search, and YouTube.
On the one hand, I think it's amazingly cool that they can take the initiative when exploring subjects that interest them. For example, Nico is really interested in astronomy, and he found some cool videos that we had fun watching together, such as the song Poor Pluto (which we're now singing all the time at our house), and another great video that gives a fun and clear explanation of the history of why Pluto isn't a planet anymore. On the other hand, there's a lot of PG+ stuff on the Internet, including (as I talked about here on Rational Moms) a weird conspiracy theory movie that Nico found all on his own and thought was a real science movie.
My personal inclination is to continue to let them explore on their own, and just check on them periodically, take an interest in what they're interested in, and offer advice/instruction based on what they discover. I know the Mormons advise keeping any computer with the Internet in a "high traffic area" of the house. Well, we have a small enough apartment that their room actually is a high traffic area.
Any other parents out there have opinions on this, or stories of what you've done with your kids? Is it irresponsible to let such small kids explore the Internet? We're the first generation of parents to even deal with this question, and I'm not sure if there's any consensus on it...