I worked for years as an engineer for Matco Tools. With our tool boxes, we were VERY clear and specific in our literature about never opening more than one drawer at a time. I know if I opened two or three particular drawers on my own Snap-On roll around, it would tip over. So it comes down to this - If the box was in good shape and on level floor, then you may of had too many drawers open and should make it right. If it was tippy to begin with, or the casters were weak and tipped, then it's not (at least entirely) your fault and you should negotiate with him.
In either case, if the drawers are just dented as you say, I think repairing them would be a fair alternative, as long as it is done properly. IMO, this would also depend on how old the box is, as if it is fairly new, I can understand why he'd want the drawers replaced versus repaired.