badmouth vs besmirch

badmouth and besmirch both are verbs.

Word	Noun	Adjective	Verb	Adverb
badmouth	No	No	Yes	No
besmirch	No	No	Yes	No

As verbs, besmirch is a hypernym of badmouth; that is, besmirch is a word with a broader meaning than badmouth:

badmouth: speak unfavorably about
besmirch: charge falsely or with malicious intent; attack the good name and reputation of someone

Other hypernyms of badmouth include asperse, calumniate, defame, denigrate, slander, smear, smirch, sully.

badmouth (verb)	besmirch (verb)
speak unfavorably about	charge falsely or with malicious intent; attack the good name and reputation of someone
	smear so as to make dirty or stained