I assume that 'dirt' here refers to information that can be used to smear someone's reputation. It's the same idea as in 'digging for dirt on someone'.
So ( also assuming... ) 'dirt at any cost' means that they want to find dirt, and are prepared to do anything to find it. I'm not sure if that is referring to the cost to the person that is looking for it, or if that cost is to the person that the dirt is on. e.g. for the 'finder' that could be risking prison for hacking phone messages, or breaking into a house to find incriminating documents. For the 'victim' that could mean destroying their career and/or relationships from having secrets exposed. Perhaps it's both?
Does that make sense?