I don't see a problem, because knowledge is not the same as information. If a piece of information is sought in a web or other electronic search, it can be likened to looking things up in books, except for some major differences, including but not limited to:
All accesses occur much more quickly.
The correct information is easier to find quickly via index searches, both because of the speed of access and because the searches can be "smart".
The information itself is vastly easier to cross-link, just as it is easier to follow such links.
Now, let's consider why a child (or anyone) would look something up in a web resource. Either it is in support of in-depth learning, e.g. an information tidbit is needed to support writing a school report or some other project, to understand a difficult passage in a text, etc.; or it is sought for basic curiosity, perhaps to understand a news item or a discussion board post.
I don't see the harm in those aspects. Speed of information access helps to offset the vast and increasing amounts of information today. Electronic tools can ease learning just as well as they can ease satisfaction of curiosity; but even for the latter, I see some potential learning benefits (e.g. certain news items become better understood, and discussions on messaging boards can become more informed over time).
ETA: If you are concerned about a child finding the answers to problems she is supposed to work herself, I suppose that is a potential issue. I wouldn't call that using the Internet as an aid, but rather as a cheating tool. Anti-cheating measures can include the use of plagiarism checkers and disallowing unrestricted access to sites where students may ask for homework help. Luckily, the answers to most math questions are not posted online, though the answers to some old chestnuts may be. Monitoring and restricting online access has to be part of the solution.
The mere fact of some reliance on a useful tool does not necessarily make it harmful either. In your Wikipedia example, the children might have grown to find Wikipedia so useful as a place for beginning research that they were at a loss in its absence, perhaps because they were unaware of other online resources that could help fill the void. For people who grew up doing homework research nearly exclusively in books, it may seem lazy to instantly conclude one is unable to to do research without electronic tools, but we also don't know the whole story in each case. In any event I don't think that's necessarily laziness talking, but rather ignorance of other research tools or methods. I can tell you that my seven-year-old child wouldn't be able to get to our town library without my help, and a request to do so based on an online outage might find me griping on Twitter (it's a good 15 minutes away, plus wait time while the child browses).
Will computer use impact the amount, or at least ratio, of memorized information? Possibly, but there is more information to deal with today than a hundred years ago; life is more complex, and the information load necessary to become tops in any field is greater too. Information is not knowledge either, and there's no indication that with proper learning supports, the electronic age in and of itself is impeding learning. Too much screen time with non-educational subjects may do that, but not simply easing access to information.
In my opinion, the main problem with internet access for children, aside from online safety, is that a lot of content is entertainment with little to no educational value, and that goes even for sites which are aimed squarely at education. My older chlid's school uses a certain e-learning tool called CompassLearning Odyssey, paying for subscriptions to it, and in my opinion that is a good example of a low-quality product: too many cartoons, too much stimulation. The same child also conned me into allowing extended play on a website called "PopTropica", on the basis that it was educational, but I thought differently after actually reviewing it. I'm leery of such things the way I avoid electronic toys for infants which reward each interaction with sounds and flashing lights: I'm afraid that my children will become conditioned to need such stimulation in order to stay focused, and I don't like anything which tends to dilute valid informational content.
On the other hand, I have found plenty of online resources that are wonderful. BBC's "Dance Mat Typing" is actually the best typing software I've found for kids, having bought several others. Khan Academy videos are fun for my kids and they do learn from them, instead of just looking information up. And when it comes to lookups, my older son uses dictionary.com all the time-- and I can't see the harm in him looking up words online versus in a paper dictionary; it just saves time and allows quick browsing to related terms, which can only help to stimulate interest in words and a broader vocabulary.
Parents will of course draw lines in different places as to what educational content is valuable and what isn't. I would just make sure that you strictly control access to online sites, including the use of "net nanny" type of software for safety reasons as well as to make sure you vet each new site for appropriateness. And take time to really find out about each site your children want to use, as well as finding new ones for them; read reviews, and actually use the sites yourself. Don't assume that just because a teacher or school recommends a site, it is valuable.