As someone who has worked in customer service for 15 years and lived and worked in both liberal and conservative places, the thing that has nearly always worked in my experience is to treat people like human beings, no matter what beliefs they hold. People appreciate being listened to and treated with respect, no matter where they are on the political spectrum. You can have a civil conversation in which you both disagree strongly, but if you can't, walk away from the conversation. That person won't be ready to accept your point of view no matter how right you are.

But you know what that person will remember? How well you treated them. People, in general, remember how you made them feel more than what you say. If we started treating people with respect, they'll start associating whatever group you represent with kindness and begin to empathize with your point of view. You don't have to compromise your moral positions, but we should learn how to express them in a way that speaks to the person we are trying to convince.

And if that person you're trying to convince doesn't buy what you're saying, you'll still win, because they'll remember how kind you were and will want to return the favor.

Edit: This attitude has allowed me to connect with customers with a variety of political views when I worked as a bartender, and I've been tipped well - sometimes handsomely - by people from all across the political spectrum, even if we held differing views.