After a whole lot of speculation from armchair lawyers following Dolphin's removal from Steam, the developers behind the popular GameCube and Wii emulator say "we do not believe that Dolphin is in any legal danger."
Back in March, the Dolphin devs announced that the emulator was coming to Steam. Some time later, Valve reached out to Nintendo over the release, and Nintendo requested that Valve block Dolphin's Steam launch, citing DMCA concerns. While Nintendo never sent an official DMCA takedown notice, the results were the same: Dolphin's Steam page was taken down, and plans for the emulator to come to the platform were abandoned.
While the developers of the emulator released only one short statement on the incident, comments from a former dev made clear that one of Nintendo's primary concerns was Dolphin's inclusion of Wii cryptographic keys - essentially a form of copy protection used on the original console. The DMCA specifically prohibits software designed to circumvent copy protection, and emulation enthusiasts soon became concerned that Dolphin's use of these keys could leave it open to big legal issues in the future.
Now, after months of silence - and some consultations with a lawyer - the Dolphin devs have issued a larger statement, and it seems they're unconcerned about facing legal action. In short, they argue that the DMCA's provision mentioned above only applies to software primarily made to break copy protection - not to something with as many other functions as Dolphin.
Additionally, the devs say "to all the armchair lawyers out there, the letter to Valve did not make any claims that we were violating a US copyright by including the Wii Common Key, as a short string of entirely random letters and numbers generated by a machine is not copyrightable under current US copyright law. If that ever changes, the world will be far too busy to think about emulation."
Since Nintendo simply made a polite request for Valve to remove Dolphin from Steam - not a formal legal action - the devs of the emulator "do not believe that Dolphin is in any legal danger." Valve is free to remove content from Steam for any reason without legal pressure to do so, and it seems that's exactly what happened here.
While everybody wants a clean, easy answer about whether or not any specific emulator is legal, US copyright law is not nearly clear enough to give us that answer. A court would have to make a decision on how the law is meant to be interpreted, and so far nobody's been willing to take an emulation case to court - outside of Sony with Bleem, where the decision turned out in favor of emulator devs. Based on the fact that Nintendo has only pursued legal action against ROM distributors, not emulator developers, you could probably guess that the company's lawyers don't believe they have a case, but even that's pure speculation.
For now, the Dolphin devs say that some features being built for the Steam version, including a 'big picture' interface that you can navigate with a controller, are still in development and will hit the standalone edition of the emulator.