Old English dumb "silent, unable to speak," from PIE *dheubh- "confusion, stupefaction, dizziness," from root *dheu- (1) "dust, mist, vapor, smoke," and related notions of "defective perception or wits."
The Old English, Old Saxon (dumb), Gothic (dumbs), and Old Norse (dumbr) forms of the word meant only "mute, speechless;" in Old High German (thumb) it meant both this and "stupid," and in Modern German this latter became the only sense. Meaning "foolish, ignorant" was occasionally in Middle English, but modern use (1823) comes from influence of German dumm. Related: dumber; dumbest.
Applied to silent contrivances, hence dumbwaiter. As a verb, in late Old English, "to become mute;" c.1600, "to make mute." To dumb (something) down is from 1933.