The dimensions at which you create/design your background art depends on:
- the camera/screen proportions and resolution
- whether you need to pan the view while animating the scene (up, down, left, right, or any combination)
- whether you need to zoom in or out of the scene while animating
- whether you have parallax background layers moving at different speeds
- the type of / art style of / details in your background art.
- vector or bitmap
If your (camera) output resolution is 1080p, and the background is a simple static non-panning, non-parralax, non-zooming set up, then create a 1920x1080 background asset in a design app. It matters not if it is bitmap or vector based (or a combination of both).
Beyond this simple case, it becomes rather more complex quite quickly. If you need to walk a character from point A to point B with a simple non-parralax one-layer scrolling background, and this walk spans two screens, you would need a 1920x2:3840 by 1080 background.
If your script requires you to zoom out while the character is walking between these two points, you need to accomodate that as well.
With vector it becomes easier, because sharpness of the art is not lost when zooming in and out (only overall detail). You would design your art at the required detail level at the highest zoom factor. But too much detail at the highest zoom level would probably look odd as well when zoomed out too much.
To answer you question in more detail, please explain more about the type of scene that you intend to animate. We could prove better and more to-the-point answers then, because your question is too broad (as you can probably tell from my answer here).
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