Unfortunately, there is no option on this popup to disable the prompt forever.
On December 18, give or take,* Adobe Systems released a security bulletin that basically says old versions of the Flash player are buggy as heck (see Flash Player update available to address security vulnerabilities).
Specifically, versions 188.8.131.52 and earlier contain nine different bugs that Adobe calls critical.
What follows are my suggestions and experiences about updating the Flash player. I read a number of articles on this topic before writing this posting and none mentioned the fact that you have to update the Flash player for both Internet Explorer and Firefox.
The two browsers use separate and independent copies of Flash.
Are you bothered with Adobe Flash Player constantly prompting you to install updates with a box that says “An update to your Adobe Flash Player is available“?