Soil by smearing. Stain with grime, or, more often, tarnish a reputation by attaching some sticky insinuation to it. Besmirch implies defilement. The harm comes not from argument but from the smear itself.

Formed with the intensifying prefix be- + smirch “a smudge, a stain,” a word of Germanic stock. The sound of smirch is already dirty; be- only spreads it further.