This topic makes me think of something that Paul Washer said at one of the Revival Conferences. He mentioned not trusting a young Christian man to be alone with his daughter because he knows (and, by implication, doesn't even trust) himself.
I think that this is the type of "skepticism" that is healthy. It isn't the utter "lack of trust" in others, but in knowing that everyone else is just as fallible and confined to humanity as we are on this side of the "glass darkly" of Eternity -- no matter how much we love or long for the Lord.
As a believer, I do not mind it when someone is skeptical of what I say. I also expect that the world -- and even some fellow believers -- might be cynical of any claims that we make.
There is an old saying that you often hear from Pentecostals which says, "The man with an experience is never at the mercy of the man with an argument." Although I understand the point in context, I think that this is a dangerous position for which we need to be extremely careful.
The Bereans were considered "noble" for searching the Scriptures to verify whether the things that Paul was preaching was true (Acts 17). Remember: This was not just some local preacher. This was the Apostle Paul -- a guy who literally raised the dead and wrote 2/3 of the New Testament.
Paul also warned about untested claims pertaining to the Gospel -- even if they came from Paul & Co. or angels (Galatians 1:8)!
I think that it is within human nature to be skeptical. In many ways, we have all been a "doubting Thomas" at one time or other. There is this sense of discovery that we feel a need to "see for ourselves" that seems innate in human beings. We believe in gravity because the ball that we throw into the air always comes back down again.
When something appears to contradict what we think we know to be true, our minds feel a need to analyze it (and not trust it until we test it for ourselves).
Speaking of gravity: When I was a child, I had a teacher who explained the Law of Gravity to our class. The teacher explained that gravity is a near-constant and will not affected by weight. He said that a pencil or pen will drop from a building at the same rate (9.8 m/s²) as a bowling ball, vacuum cleaner or any other much heavier object.
As a 2nd grade little boy, I didn't believe the teacher. After class, I explained that this just didn't make sense to me. The following week, the teacher took us outside and "verified" (via a test) that this was true. We had a maintenance worker drop bowling balls and a pencil at the same time from various heights (roof and a lift). I was astonished...because I had seen with my own eyes what he was saying.
As I grew, the curiosity about WHY this was true led me to research it further as I grew in both age and knowledge of physics. During a university Physics course, I was surprised that there were some students who were "shocked" to learn this (because they hadn't learned or studied it prior to college).
At the same time, the truths and works of God aren't bound by the earthly laws that God ordained for Creation.
I always thought that one of the interesting things in the story of Lazarus wasn't just that Jesus raised him from the dead (after three days), but that whatever unnamed affliction led to his death didn't apparently continue to plague him after he was raised back to life.
I suppose that we should very much understand the skepticism that the world shows toward the Word of God. Moreover, there are believers who struggle with faith when so much of the world literally screams ideas that are contrary to what is contained within God's Word.
When I speak with atheists (or even those who struggle with faith), I always let them know that I was once just as skeptical as they are. I let them know that I didn't believe in God until just prior to my conversion. I also acknowledge the apparent contradictions between the claims of science and the claims of God's Word.
This is why I think that a testimony is important to this lost world. The world and all of those who struggle with faith need to know that we were once just like them. They can identify with the skepticism that we once held and may be interested in what led us down the narrow road of faith. Thus, there is power in the word of our testimony!
The world stereotypes believers. It is easy to see why. They stereotype believers as the ignorant people of this world who are willing to take things by faith rather than by testing. To the world, Christians are a gullible bunch who are willing to accept anything that we are instructed to. They point at the large, eager congregations that follow some very wild and wacky preachers who are motivated by fame, money and success.
Yet, when the people of this world meet us, do they feel that we are gullible? Do they see us as people willing to "test everything?" Do they see us as having come to a "born again" experience that "changed everything" (in terms of our thinking)?
While this is mostly in consideration of non-believers, there are also many believers who haven't been planted on "good soil" and struggle with taking root in shallow soil or the words and wisdom of this world that can choke their faith shortly after it has begun.
Whenever I speak with unbelievers or atheists, I often encourage them -- even with doubt -- to speak to God when they are alone. I urge them to use as much personal openness to ask God to "reveal" himself to them. I even tell them to admit their skepticism to God when they speak. I explained that God doesn't always answer them by "fire" (or the miraculous), but that He knows how to grab the heart (and attention) of those who sincerely want to know Him.