In my experience, and based on my professional photography skills, the short answer is to get a better scanner. I'm not trying to be crass, but I think I'm correct.
I had a real nice stand-alone scanner (not part of a multi-function printer, etc), and my girlfriend has a professional photo scanner. Both have a long "depth of field". My Brother All-In-One Printer/Scanner etc. has a very short depth of field.
Depth of field means how far your scanner lens can "see". A larger depth of field means objects that are farther away from the glass will still be in focus (like the card in a BGS case - it's not pressed right up on the glass like in a toploader). A shorter depth of field means that only things that are right on the scanner glass will be in focus and not blurry (like a "bare" card, a card in a toploader, or a sheet of paper).
My new Brother All-In-One has such a short depth of field that even sometimes patches that are not right at the front of a card come out a little blurry (like a Scripted Swatches card). My old scanner, I could scan an action figure and most of it would be in focus - it seemed to have a depth of field of almost 1/2 an inch...
You could sharpen the image in something like PhotoShop (which I do sometimes), but even then - if you start with a blurry scan, there's not much that can be done to correct it.
Higher end scanners seem to have better optics - better lenses - and therefore have an increased depth of field, allowing greater scan flexibility.
Sorry for the technical lesson here. If anyone can refute what I'm saying or give better tips, let me know. I'm not sure the color of the background has anything to do with the quality of the scan - but if someone has a better tip, I'll use it too!
