Contrary to common belief - halacha prescribes punishment - even capital punishment based on circumstantial evidence and presumption (chazaka). Even strong rumors are enough to give punishment
Kiddushin (80a): This was taught only in respect of Sanctities of the border, but not in respect of genealogy. But R. Johanan maintained: Even in respect of genealogy. Now, R. Johanan is in accord with his view [elsewhere]. For R. Hiyya b. Abba said in R. Johanan's name: We flagellate on the strength of presumption, we stone and burn on the strength of presumption, but we do not burn terumah on the strength of presumption. We flagellate on the strength of presumption, as Rab Judah. For Rab Judah said: If a woman was presumed a niddah by her neighbours, her husband is flagellated on her account as a niddah. We stone and burn on the strength of presumption, as Rabbah son of R. Huna. For Rabbah son of R. Huna said: If a man, woman, boy and girl lived in a house [together], they are stoned and burnt on each other's account.6 R. Simeon b. Pazzi said in R. Joshua b. Levi's name on Bar Kappara's authority: It once happened that a woman came to Jerusalem carrying an infant on her back; she brought him up and he had intercourse with her, whereupon they were brought before Beth din and stoned. Not because he was definitely her son, but because he clung to her.
שולחן ערוך (אבן העזר יט:א): מי שהוחזק בשאר בשר, דנין על פי חזקה זו אף על פי שאין שם ראיה ברורה שזה קרוב, ומלקין וחונקין וסוקלין על חזקה זו.
Aruch HaShulchan (C.M. 2:1): Even though the Jewish court does not judge cases involving capital punishment or flogging or fines outside of Israel – but if the beis din sees that the times require it - because there is a breakdown in law and order – then it is permitted. Everything depends on the judgment of how serious the problem is. Not only can the court judge these cases when there is community lawlessness but even when a single individual sins it is permitted to punish him if they think it is necessary - as long as they do it for pure motivation. This ability to judge these cases in emergency situations is even when there are not valid witnesses but only a reasonable basis that it is true and constant rumors are heard when there are no enemies who would create them
(Rambam (Hilchos Sanhedrin 24:5): A beis din – in all places and for all times - has the power to flog a person about whom there are rumors of bad behavior and the people and complaining that he has transgressed prohibited sexual relations. This is only when the rumors are unceasing as we have explained and also that he doesn’t have any known enemies who would slander him. ….
Yam shel Shlomo (Yevamos 10:20):… Similarly the beis din has the right to flog a person who has a bad reputation and there are constant rumors that he is involved in improper sexual relations. However this is only when he doesn’t have enemies that would spread lies about him
Kiddushin (81a): Rav said that we flog a person because of an evil rumor about him
Meiri(Kiddushin 81a): All those who act with inappropriate frivolity until people are constantly complaining about him that he is a sinner is to be flogged based on his activity which leads people to give him a bad reputation – even though no one has given formal testimony that he has sinned… Whoever acts in a way that causes rumors that he is a sinner has violated a negative commandment from Tradition.