When you said "I have been happily using Premier and Encore CS6 on my Mac OS X (Version 10.9)," I took that to mean that you had already been using it on 10.9. Some Mavericks users are able to open Encore and run into problems with other issues.
Is the rest of your adobe software working? If not, look to those forums (e.g. Premiere Pro or After Effects) where there are specific issues being addressed.
If yes, are you trying to open a project? Or just to open encore without opening a project until you get to that screen?