Looking at the base class (Popup) I see that there is a property called key which seems to correspond to a member of dojo/keys for this very purpose. Sadly, all the relevant code is commented out.
Has this functionality been deprecated, or can we expect to see it implemented in a future release?
The likelihood is that the code was never completed or had some browser compatibility issue. You can try and un-comment the relevant code and test in your app deployment environment to see if it works for you. Other than that you can call support and log an enhancement request for this feature to be completed. Normally esri development staff does not monitor or respond to threads like this so you are unlikely to get a better answer.