Christians should care about how what they say affects people.
However, Christians shouldn’t be politically correct when that means abiding by a political agenda that is anti-Christian.
What should a Christian think of being politically correct when many of the people who demand it so adamantly seem to be anti-Christian? What needs to be understood about the current climate surrounding the issue is that many of the proponents of politically correct rhetoric use the term to portray words and ideas they personally do not agree with as politically incorrect. There is a difference between what political correctness should mean and how the term is used for ulterior motives.
A practicing Christian should be politically correct in the sense that they should care about how what they say affects others. However, Christians shouldn’t be politically correct when that means abiding by a political agenda that is anti-Christian. For example, when a Christian discusses their faith at school or work, they shouldn’t think that what they believe is wrong only because they are told that what they are saying is not politically correct.
Here is a rule of thumb. When a Christian talks to someone, they should think of how they would feel if they were in the other person’s shoes during that discussion. A Christian should choose their words based on that understanding, but that doesn’t mean that a Christian should only say what the other person wants to hear or that it is wrong to disagree with that person.