害する
がいする · gaisuru · JLPT N1 · verb-suru
to harm
害 = harm + する = do: to do harm, which also covers offending someone
害 (gai) is the noun for harm or injury, and する (suru) turns it into an action, so 害する literally means to do harm. The same verb covers harming someone's feelings, which is the offend sense.
害 — 宀 roof + 丯 (phonetic) + 口 mouth
Originally an ideograph showing harm spoken under a roof, with 宀 (roof/house) on top, a phonetic element 丯 in the middle, and 口 (mouth) below suggesting harmful speech. Over time it shifted from 'injury caused by words' to the broader senses of 'harm,' 'injure,' and 'fear.'