You should ideally have a default resolution of 1920x1080, 1280x720 or 640x480, which are the 3 typical hd resolutions
Hey, I don't know that 640x480 is a HD resolution... ;-) ;-)
Btw: painting a background much bigger temps to make too much details. After resizing the picture seems overloaded. You have to paint as you should be in the final size after downsizing, that means: making all lines thicker, all colours stronger, enough space between lines and points, less grey shadings and so on. It's like making a stamp: the Mona Lisa on 2x3 cm should be a different Mona Lisa than on 200x300 cm...