Does the Bible say we should not care what others think about us? No, but it does say we must seek to please God over people.
As every Christian matures, one of the things the Holy Spirit will surely convict us of is seeking the praise of man over God. In our sinful nature, we care about what others think of us for the wrong reasons.
Caring What Others Think of You Can Be Expressed Wrongly in Two Ways
When we seek the praise of people, it can be expressed in many different ways. The two ends of the spectrum are obsessive pursuit and obsessive avoidance. One person may do anything to be praised by people, while another person will do anything not to be seen by people, but both can have the same root issue – caring too much about what people think of them.