There are three kinds of doubt: The reasonable, the unreasonable, and the reasonably unreasonable. Sometimes, doubt just makes sense. You hear something that doesn't fit your current perception of reality, you don't see much evidence to accept it, you have good reason to doubt, and so your doubt is reasonable. Sometimes, doubt makes no sense. You might even be actively changing your criteria for acceptance just to avoid accepting it, and so your doubt is unreasonable. Sometimes, you have good reason to accept something, and may even actually accept it, but you just have this annoying feeling that, maybe, your belief might be wrong. This is reasonably unreasonable.
In the Bible, we see all sorts of doubt. Indeed, one of the purposes of the Bible is to remove doubt. But it won't always be enough. It is, after all, only a book, and a rather large one at that. Therefore, as Christians, we are supposed to respond mercifully to doubt in order to address it. And God gives us examples, for He, Himself, encountered many doubters.
First, let us consider Nathanael (John 1:43-51). When Philip told Nathanael the Messiah had been found, Nathanael doubted. He was told Jesus was from Nazareth, and he scoffed, and yet Philip beckoned him to see for himself. And Jesus saw Nathanael coming, and said of him "Behold, an Israelite indeed, in whom is no deceit!" Note Jesus' response here. Behold this doubter! When he was told of Jesus, he scoffed, and yet rather than rebuke his doubt, he looked him right in the eyes and declared his integrity. Sometimes, there are doubters who are very honest.
Second, of course we have the Pharisees. The Pharisees were so unbelievably unreasonable that when Jesus did His miracles, they blamed them on Satan. And Jesus, knowing their hardened hearts, rebuked them sternly. He called them hypocrites, pointing out that they were able to piece together the signs of the weather, yet were unable to discern the signs of the times (Matthew 16:1-3). Jesus gave them plenty of evidence. His miracles, His fulfilment of prophecy, the fact that He knew scripture better than any of them ever could, and yet the best they were willing to grant is that He was doing miracles... by power of Satan (Matthew 12:24; Mark 3:22). And so Jesus rebuked them strongly, and told them "no sign will be given except the sign of Jonah". (Matthew 16:4).
There comes a point when you have to just accept that a doubter is going to doubt. There are people out there who will even admit that if all the evidence pointed to God, they would reject Him. At this point, doubt is inevitable. It is acceptable to give them evidence, yet they will reject it, and so it is better for you to just drop it and leave.
Third, let us consider John the Baptist. John the Baptist arguably had more evidence than anyone that Jesus was the Messiah. Nevertheless, as he was in prison awaiting death, he sent a letter to Christ: "Are You the Coming One, or do we look for another?" (Matthew 11:3). And Jesus did not rebuke him! Here we have an unreasonable doubter, but he is not unreasonable in the sense of the others. He is unreasonable because he has reason to be so. He had certain expectations about what the Messiah would do that certainly didn't include him being executed. Yet here he was, in prison awaiting death, and so naturally he wondered. Instead of rebuking him, Jesus did a number of miracles and sent back word to John: "Go and tell John the things which you hear and see: The blind see and the lame walk; the lepers are cleansed and the deaf hear; the dead are raised up and the poor have the gospel preached to them. And blessed is he who is not offended because of Me." (Matthew 11:4-6).
And so rather than rebuking John for his doubt, He encouraged him in spite of it. "Yes John, you doubt, yet here are the reasons you shouldn't." Jesus had the right to be ticked because John doubted, but instead, he acted as if John had the right to doubt.
Through these three examples, we see that doubt is not something we should rage about, as many of us do. We don't need the unpleasantries we see online, or in our churches, or even in our social groups. If someone doubts, Jude tells us very simply that we are to be merciful to them. In fact, as far as I am aware, this is the closest the Bible comes to flat out saying "hate the sin, love the sinner". In effect, Jude says "hate the doubt, love the doubter". My brethren, love the doubter. In fact, I am going to go one step further and remind you that sometimes, you will be the doubter. I know I certainly have. This whole ministry is a result of the fact that God was merciful to me as a doubter. And you could easily become doubters too. So be merciful to those who doubt, and pray that when you doubt, others will have mercy on you. The Church grows, and gains strength, only when we know who our enemies are, and more importantly, who they are not.