Drop shadow for Captions? (Premiere Pro 14.9 and earlier)

Highlight all the captions in the timeline, go to the essential graphics tab, look for background, then select both the color and opaque, translucent, or transparent at the bottom of the drop menu. You have to select both for a shadow box to show up! I apologize for the poor quality of photos as they were taken of the computer screen with my phone

Simple answer is no, you cannot add drop shadows to captions within the Captions option in Premiere Pro.

You can manually create titles with that effect and use them as subtitles in your footage. However, you cannot export captions as separate text files (srt), as drop shadow is not part of any text format: it's a design choice.

The only way to export the titles separately would be to export them out as a separate layer (on a transparent canvas) with transparancy enabled. Either as a complete video that matches the length of your destination sequence or as separate images files. With the latter, you would need some form of guidance to tell the player when which image should be displayed. This is typical for .idx and .sub formatting.