Is there a recommended file size and type for the banner files used on responsive skins in the banners widget?

There is not a recommended or required size.  However, the banner images will respond to the browser width and shrink or grow accordingly. We do suggest that you look at the uploaded images on the largest screen size - such as a monitor for a desktop computer.  Make sure they look good on the larger screen - checking to make sure the images are not pixelated.  Also, make sure that their byte size it not far over 200kb.
We recommend that you only use .jpg images that are optimized for the web. Any other images will be too large for mobile devices.