You have multiple problems to solve here. Your background has the same type of constrain as most textures and 2d assets. Your textures (and other assets) have to adapt to hardware with:
- different resolutions
- different aspect ratios
- different pixel densities
- and different performances
In this case you will have to make sure that your game looks good on the best hardware and is not a pile of crap on the rest you have to make compromises. First, you will have to ensure that you have assets for the best hardware and the worst (think pixel densities).
So make your assets for high resolution screens. Make sure you support the highest resolution available.
Also make sure your background has room for different aspect ratios. Therefore make it so you have room on the sides to expand on larger or higher screens.
Ensure that you use the information about pixel density screen size and other indicators to avoid displaying the wrong assets. The resolution is not necessarily the best way to decide which asset to use. For example text must be readable on high density screens and it should be smaller on low density large screens (tablets) so if you have some details to show in your background you might want to dissociate the details the user must see from the background and ensure users can see them on all devices (density/size/form factor).
And finally to accommodate most of the form-factors and use the lowest footprint you might want to divide the largest resources into tiles or crop them at runtime. To reduce processing time at initialization you might also want to add assets for lower density and lower quality screens.