To stumble is to trip, but not necessarily fall down. To "stumble around" is to be unsteady on your feet, possibly bumping into things, or to be unbalanced. While the phrase is used in your sentence with the idea of alcohol causing it you'll also often hear "stumbling around in the dark" which means that someone is having difficulty moving through a dark area, bumping into things, knocking things over, tripping over things, etc. (ES: dando tumbos)
Alcoholics don't run in my family, they stumble around and break shit.
Songs that contains the expression