The answer is not simple. Some biblical texts seem to say no. Others, yes.
For example, Jesus warned us: "Woe to you when all men speak well of you" (Luke 6:26). And his own enemies saw in him an indifference to what others thought: "Teacher, we know that you are true, and care for no man; for you do not regard the position of men but truly teach the way of God" (Mark 12:14). Paul said that if he tried to please men he would no longer be a servant of Christ: "Am I now seeking the favor of men, or of God? Or am I striving to please men? If I were still trying to please men, I would not be a bond-servant of Christ" (Galatians 1:10). "As we have been approved by God to be entrusted with the gospel, so we speak, not as pleasing men, but God who examines our hearts" (1 Thessalonians 2:4).
On the other hand, Proverbs 22:1 says, "A good name is to be more desired than great wealth, favor is better than silver and gold." And Paul was vigilant that he not be discredited in his handling of money for the poor: "[We are] taking precaution so that no one will discredit us in our administration of this generous gift; for we have regard for what is honorable, not only in the sight of the Lord, but also in the sight of men" (2 Corinthians 8:20-21). It mattered what men thought. He taught the Roman church, "Now we who are strong ought . . . not just please ourselves. Each of us is to please his neighbor for his good, to his edification" (Romans 15:1-2). And he taught that one of the qualifications for elders is that they must be "above reproach" (1 Timothy 3:2), including among unbelievers: "He must have a good reputation with those outside the church, so that he will not fall into reproach and the snare of the devil" (1 Timothy 3:7).
Similarly Peter charged us to care about what outsiders thought: "Keep your behavior excellent among the Gentiles, so that in the thing in which they slander you as evildoers, they may because of your good deeds, as they observe them, glorify God in the day of visitation" (1 Peter 2:12).
Question: How is the tension between these two groups of passages to be resolved? Answer: By realizing that our aim in life is for "Christ to be magnified in our bodies whether by life or by death" (Philippians 1:19-20). In other words, with Paul, we do care - really care - about what others think of Christ. Their salvation hangs on what they think of Christ. And our lives are to display his truth and beauty. So we must care what others think of us as representative of Christ. Love demands it.
But notice where the accent falls: not on our value or our excellence or our virtue or our power and wisdom. It falls on whether Christ is honored by the way people think of us. Does Christ look great because of the way we live? It matters to us whether he does.
Again notice a crucial distinction: the litmus test of our faithfully displaying the truth and beauty of Christ in our lives is not the opinion of others. We want them to see Christ in us and love him (and thus incidentally approve of us). But we know they may be blind to Christ and resistant to Christ. So they may think of us just what they thought of him. "If they have called the head of the house Beelzebul, how much more will they malign the members of his household!" (Matthew 10:25). Jesus wanted men to admire him and trust him. But he did not change who he was in order to win their approval. Nor can we.
Yes, we want people to see us with approval when we are displaying that Jesus is infinitely valuable to us, but we dare not make the opinion of others the measure of our faithfulness. They may be blind and resistant to truth. Then the reproach we bear is no sign of our unfaithfulness or lack of love.
May God give us wisdom and love and courage to please and not to please as we hold fast to Christ our treasure.