Why are Christians feared or hated? For example after Saul had his Damascus Road experience, he was doubted by other disciples until Barnabus spoke out on his behalf. Why would they fear a fellow Christian? In general, why are Christians perceived negatively by the world, and how might we change that perspective?
First, Christians are often maligned for their faith. The Bible says that Christians will be hated by the world, because we are not "of the world."
Jesus taught this most forcefully in John 15:18:
If the world hates you, you know that it has hated Me before it hated you. If you were of the world, the world would love its own; but because you are not of the world, but I chose you out of the wrld, because of this the world hates you. Remember the word that I said to you, ‘A slave is not greater than his master.’ If they persecuted Me, they will also persecute you; if they kept My word, they will keep yours also. But all these things they will do to you for My name’s sake, because they do not know the One who sent Me.
In other words, it is natural and expected that Christians should be mocked and rejected by the unbelieving world, just as Jesus Himself was. The Bible also teaches us that we should take some measure of comfort in that rejection, as Jesus said in Matthew 5:11-12:
Blessed are you when people insult you and persecute you, and falsely say all kinds of evil against you because of Me. Rejoice and be glad, for your reward in heaven is great; for in the same way they persecuted the prophets who were before you.
It's also important to remember that we should not seek the world's approval or try to become like the world so that they might accept us, as Jesus taught in Matthew 5:13-16:
You are the salt of the earth; but if the salt has become tasteless, how can it be made salty again? It is no longer good for anything, except to be thrown out and trampled under foot by men. You are the light of the world. A city set on a hill cannot be hidden; nor does anyone light a lamp and put it under a basket, but on the lampstand, and it gives light to all who are in the house. Let your light shine before men in such a way that they may see your good works, and glorify your Father who is in heaven.
So all Christians should understand and accept that our new life in Christ has made us different from the rest of the world to such a degree that reconciliation isn't possible nor desireable. Unbelievers will alienate us and even hate us at times for Jesus' sake. The Bible says this is the natural and intended state of affairs, since our difference in the world is our witness.
While we wish to be at peace with all men so far as it depends upon us (Romans 12:18), we also remember that we cannot seek the approval of men over the approval of God:
Woe to you when all men speak well of you, for their fathers used to treat the false prophets in the same way. - Luke 6:26
Having said that, we can certainly make reasonable efforts to avoid conflict with others, and even to work and live closely with unbelievers. Paul himself did this routinely, and he taught that he was willing to become "all things to all men" in an effort to win some to Christ. Obviously, Paul didn't compromise His closely-held Christian beliefs or morals, but he was willing to forgo virtually everything else if it might increase his chances of winning men to Christ. Paul describes it this way in 1Cor 9:20-23:
To the Jews I became as a Jew, so that I might win Jews; to those who are under the Law, as under the Law though not being myself under the Law, so that I might win those who are under the Law; to those who are without law, as without law, though not being without the law of God but under the law of Christ, so that I might win those who are without law. To the weak I became weak, that I might win the weak; I have become all things to all men, so that I may by all means save some. I do all things for the sake of the gospel, so that I may become a fellow partaker of it.
We should seek to reduce the potential conflict we may experience with unbelievers by approaching them without pretense or judgment. Christians should avoid showing pride or hautiness from our faith, and we can't carry a superior attitude among those who are without Christ, since their sins are no greater than our own and our faith is a gift, as Paul taught:
Quite right, they were broken off for their unbelief, but you stand by your faith. Do not be conceited, but fear; - Rom 11:20
Showing genuinely kindness and sincere interest in the the lives and circumstances of others will go a long way toward opening doors of conversation for the sake of Gospel. Nevertheless, a Christian must never forget that strife and rejection and hatred will come sooner or later, for the Lord has said it is inevitable.
Finally, the example you offered of Saul is probably not relevant to your question, since the circumstance of Saul's arrival in Damascus were quite unique. The people who met him in Damascus were Christians, not unbelievers, and they were frightened and wary of Saul not because he was a Christian. On the contrary, they were wary because they suspected he wasn't a true Christian.
At the time of Saul's conversion and delivery into Damascus, the early church only knew Saul as the man who persecuted and murdered Christians, so when they learned that Saul was staying with Christians in their city, no one knew if they could trust him. Their reluctance to accept Saul was not because they didn't know how to accept Christians.
Rather, they were Christians who didn't know whether to trust Saul, because they couldn't understand how God had changed Saul so suddenly. Only after time did everyone come to accept that Saul was truly now Paul, a man who knew and followed Jesus as Messiah.