Should Christians Care What Others Think?

Yes. Many times Yes.

Here’s what Paul says:

Gal. 6:10 So then, whenever we have an opportunity, let us work for the good of all, and especially for those of the family of faith.

1Th. 3:12 And may the Lord make you increase and abound in love for one another and for all, just as we abound in love for you.

1Th. 4:11 to aspire to live quietly, to mind your own affairs, and to work with your hands, as we directed you,  12 so that you may behave properly toward outsiders and be dependent on no one.

1Th. 5:15 See that none of you repays evil for evil, but always seek to do good to one another and to all.

1Cor. 10:32 Give no offense to Jews or to Greeks or to the church of God,

1Cor. 14:22 Tongues, then, are a sign not for believers but for unbelievers, while prophecy is not for unbelievers but for believers.  23 If, therefore, the whole church comes together and all speak in tongues, and outsiders or unbelievers enter, will they not say that you are out of your mind?  24 But if all prophesy, an unbeliever or outsider who enters is reproved by all and called to account by all.  25 After the secrets of the unbeliever’s heart are disclosed, that person will bow down before God and worship him, declaring, “God is really among you.”

2Cor. 8:21 for we intend to do what is right not only in the Lord’s sight but also in the sight of others.

Phil. 4:5 Let your gentleness be known to everyone. The Lord is near.

Rom. 12:14   Bless those who persecute you; bless and do not curse them.  15 Rejoice with those who rejoice, weep with those who weep.  16 Live in harmony with one another; do not be haughty, but associate with the lowly; do not claim to be wiser than you are.  17 Do not repay anyone evil for evil, but take thought for what is noble in the sight of all.  18 If it is possible, so far as it depends on you, live peaceably with all.  19 Beloved, never avenge yourselves, but leave room for the wrath of God; for it is written, “Vengeance is mine, I will repay, says the Lord.”  20 No, “if your enemies are hungry, feed them; if they are thirsty, give them something to drink; for by doing this you will heap burning coals on their heads.”  21 Do not be overcome by evil, but overcome evil with good.