Is caring about reputation related to pride?
Your example can be what gets your reputation. You want to be a good example, so your reputation can be good because of your good example helping other people. But, of course, you can not control how people see you. So, be God's good example, so at least people see what is good, even if they take it the wrong way.
I do my best to maintain good reputation but it can becoming worrying.
It depends on what is supposed to give you a good reputation. I have been brought up where I could be shamed if I did not conform to people who were selfish. So, if I conformed to them so they said nice things about me, that might not mean I had a reputation that was truly good.
In any case, we should not do things only because of fear. Jesus has us doing what is good . . . because of love.
Is it alright for Christians not to care about reputation?
We need to care about our example. And we care about how people react > I mean, we pray for God to bless people to benefit from our good example.
I now think of this > you can be famous, and be accepted because you have somehow pleased a lot of selfish people. But the best way to be famous is because you have been kind to some person, and have helped him or her to find out how to love. The fame you have with that one person is greater than the fame of pleasing billions of selfish people.
"quality, not quantity"