There has been an ongoing debate about whether leaving your wi-fi router open (not secured by a password) makes you liable if someone else uses your router to download unlicensed copyrighted material. For instance, some copyright owners are calling it negligence to leave your wi-fi router open.
However, many others have put together cogent arguments that leaving your wi-fi router open does not create any liability. For instance, the good folks over at the Electronic Frontier Foundation wrote a great piece that discusses several reasons why open routers should not make their owners liable. Several others have also written similar articles. You can read one here and here.
The bad news is that none of these cases have been pushed forward enough that a U.S. Court has been asked to render a legally binding decision on this issue. The good news that at least one court has issued an opinion on the matter, and it makes common sense. In Finland, a court has held that using a router that is not secured by a password does not give rise to liability for copyright infringement of third parties, i.e., people who use your wi-fi without your permission.
This decision is a huge victory for the general public, at least in Finland. Conceptually, the same should be, and hopefully will be, the law in the U.S. To hold otherwise would prevent so many open wi-fi establishments from offering free internet access to their patrons. For example, most fast-food chains, like McDonald's, provide free wi-fi access to their customers to use while eating. Very many hotels offer free wi-fi access to their guests while staying at the hotel or sitting in the lobby. Most travelers, and especially business travelers, have more need for wi-fi in their rooms than a telephone.
And even more fundamentally, you would not be liable if someone entered your house without permission and committed a crime, even if you left your door unlocked. There is no reason that a different standard should apply just because the recording and entertainment industries have found a new way to extort money out of the ordinary public.