It is essential to use different slider images for desktop and mobiles. Because, the desktop slider image will look very small in terms of length. Any text over slider will have very little space. Hence, for mobile we must use an entirely different image.
For mobile I use – 500px x 375px. While writing media queries. You should also consider the landscape resolution (will be discussed soon).
The resolution – 960px x 540px seems like a good fit on a tablet.
For desktop whatever looks good to your eyes should do it. In one of my recent project, I’ve used – 1900px x 600px