GoPro Hero 3 + Adobe Premiere Elements 14 black border
You need to select a Custom size, because 1280x960 is not an HD ratio (16:9) but something else (4:3).  The black borders are the program's way of telling you "the image aspect ratio doesn't fit the target aspect ratio, so I'm going to shrink it to fit".  YouTube, for example, will also do this to your video, as will any HDTV.  So unless you have a 4:3 end-to-end pipeline (which you might have if you are "projecting" the video onto a virtual 4:3 screen within some animation), something somewhere is going to have to crop or fill your image in ways you don't like.  The best alternative is for you to make the decision how you are going to fit it into a 16:9 world.