1. to speak negatively about something or someone; stain a reputation
don't drag my name through the mud
If you get mud on something it becomes dirty. Similarly, if you 'drag someone's name through the mud' it means you dirty their reputation. If someone is 'dragged through the mud' the suggestion is that their name, reputation or public face is disgraced or called into question, usually unfairly. This phrase is often used in reference to public figures such as celebrities or politicians. You can hear this used in social and professional contexts.