Should you care about what others think? Though some disagree with this, the short answer is... yes, sometimes.
It is popular to have a "F*** YOU" attitude toward others and do whatever you want with no regard for what others think of you. It feels... independent, strong, though probably a cover for deep feelings of rejection. Though there is a grain of wisdom found in this, minus the "screw you" attitude, it is missing one important thing: reputation really does matter. This includes our own reputation but also the reputation of others relating to our actions.
For example, I don't post personal things on Facebook. There are a few reasons for this. One, I don't want to present myself as an immature drama queen (because reputation is important to me). Two, I have children who are on my Facebook, and friends of those children, and other people who otherwise personally know or know of some of the situations and individuals I would be writing about. I would not want to hurt them or embarass them or tarnish their reputation in anyway.
The bottom line is that certain actions make us (and others) look like fools while certain actions (or the absense of certain actions) shows us to be conscientious and stable individuals, and it is good to recognize this. A good reputation is something worth pursuing.
"A good name is to be chosen rather than great riches, and favor is better than silver or gold." (Prov 22:11)
But why? What makes a good reputation such a good thing? Here are three big reasons I can think of.
1. Devotional: Your actions before others to a large degree reflect upon what (or Whom) you claim to believe in. You bring glory to the God you claim to know by being known as a person of integrity and love. Conversely, you shame God's name when you act like a fool or a liar before others.
2. Practical: In general, acting with integrity and love leads to a good reputation, which gains the respect of others. You become a person "looked to" and listened to, trusted. This means you have more influence to do good, to help and love others, and to pass on good values to others. People are more likely to partner with you, either when you are helping them or when you need their help. Conversely, people are less likely to rely upon you in the long term if your actions show you to be a fool, regardless of how "wise" you may sound at first. They are also less likely to give you help when you truly need it.
3. Social: With a good reputation you become a jeweled crown to your family and friends, someone they can be proud of knowing and belonging to. You give glory and a level of status to those closest to you. Conversely, you set a dark cloud over your family and friends when you are known for being selfish, immature, deceitful, or impulsive and foolish.
As a general rule of thumb, act in such a way that your children would not be ashamed of you if they found out what you were doing. There are a handful of things I've done in the past that do not comply with this rule, but I still believe it is a good one. It is one I take to heart.
But of course, there is a sense in which we ought NOT to care what others think. When it comes to doing what is right and sticking to what you know to be right and true, it is better to stand your ground than crumple under the weight of others' opinions. When it comes to doing what God sent you into this world to do, let them whine and complain and insult you. You'll only be joining the long line of people before you who stuck with the mission and identity God gave them and suffered for it.