How should a person act, according to Islamic Sharia, upon receiving an anonymous message informing them of an illicit relationship between their sister and someone they know?

We advise you to verify and not to tell anyone. Do not think ill of your sister or the other person without clear proof, because thinking ill of someone whose outward appearance is righteous is forbidden, as Allah says: "Avoid much [negative] assumption. Indeed, some assumption is sin." Be keen on guiding your sister and preserving her honor with an enjoyable and beneficial program, and observe her discreetly. If you notice anything, then advise her. If the man mentioned is acceptable to you as a husband for her, then propose to him, or to someone else whose religion and character you approve of, that he marry her. Indeed, Omar offered his daughter Hafsa to Abu Bakr and Uthman, and the righteous man offered his daughter to Moses (peace be upon him).