How can I respond in a faithful, caring way to people I completely disagree with?
Disagreements are a difficult part of our human experience, but for Christians and for the church, they have always been with us. We need only look at Paul’s letter to the Galatians or the writings of fourth-century Christians about the relationship between Jesus and God during the First Council of Nicaea to see that Christians have disagreed strongly with each other about many things throughout history. So, your question is a good one; “How do I respond to, or affirm someone with whom I disagree?”
One suggestion I will offer is to remember that the view that someone holds that differs from yours is not the only thing that determines who that person is. Regardless of whether you disagree with who that person votes for, their views on social issues, their faith or absence of belief, or any other concern, you may still be alike in more ways than you are different. In a 2009 TED talk, Nigerian writer Chimamanda Ngozi Adichie spoke about the danger of a single story, or the danger of allowing only one part of a person’s identity frame our perception of that person. Here is a link to that TED talk, and I highly recommend it.
If you are someone who believes in the mission of the church, Michael Kinnamon and Jan Linn write about the implications of covenant relationship in their book Disciples: Who We Are and What Holds Us Together. They write,
This means, of course, that a church built on covenant must accept, even value, a diversity of perspective. The commitment we make is not always to agree, but to walk together as we attempt to discern where God would have us go. If the church is a purely voluntary association, then when we disagree we can simply take our marbles and leave. If, however, the church is a covenantal society constituted by God’s initiative, then we stay at the table, especially when it is tough to do so.” 
These are challenging words from Kinnamon and Linn, but for me they describe part of the “heavy lifting” of discipleship.