A way to both brace and damp the stand's walls would be:
1- Install plywood strips (or dowels) about 3" apart across the width and depth, and 6" apart vertically. There will therefore be four braces every 6" of the stand's height; two running side-to-side, two front-to-back.
2- Install four braces running top-to-bottom, on the outside of and butted up against the horizontal braces. If you want to get really nutty, glue the braces together at every juncture (there will be four points at every brace level where the braces cross one another). This will create the ultimate form of bracing, a matrix.
3- Install plywood or MDF planks on the inside of the matrix the braces have formed, creating an inner box.
4- Fill the void between the inner box and the outer stand walls with sand.
The stand doesn't need braces 3" apart, but if you put only one in the middle, front-to-back, side-to-side, and top-to-bottom, there is no way to build an inner box to create a void for the sand. Of course, you could skip the inner box and just put in one brace left-to-right and front-to-back every 6" of the stand's height, and one top-to-bottom, and fill the entire stand with sand. Either way, the stand should be deader than a doornail!