On what computer operating system is your Premiere Elements 13 running? Are you working with tryout or purchased product?
Have you updated 13 to 13.1 update using an opened project's Help Menu/Update. If not, please do so.
I have not seen this in Premiere Elements 13/13.1 and have not seen this problem reported previously.
What is your end product - DVD-VIDEO standard or widescreen on DVD disc or AVCHD format on DVD disc?
What are the properties of the source media, especially those files to which the Shake Stabilizer has been applied?
What is your project preset? Are you using the Quick or Detailed mode for the Shake Stabilizer?
Let us start here and then decide what next.
Any questions or need clarification, please ask.
Have you resolved your issue yet?
Tonight I went through several run throughs and could not reproduce your Shake Stabilizer issue.
Therefore when you do reply, please include all details possible starting from the import of the video to the time to hit the burn button in the burn dialog.
If you could post a jpeg screen capture of the message being displayed in the video playback, that would be very helpful.
Important question...What is the exact wording on the purple ban? Does it actually say "Stabilizing" rather than "Click here to stabilize"