This is a threat by one party that what the other party seeks in the negotiation will not play well in the court of peer or public opinion. This is different from a warning, as it is made to tactically influence another and works well with social pressure acting on a party.
For example, you may hear, “Congressman, if you don’t screen immigrants at the border, your constituents will think you are creating a security risk.” Or, “your public won’t listen to your music if you do this, so do what I tell you.”
The answer to this is truth. Advocate for your position and be indifferent to those who threaten. Reputable public relations firms may be beneficial, but the target ultimately must take responsibility for their stance.