Explanations of Visions
My first inclination is to say that no one should believe anything based on a vision alone (regardless of who experiences it). According to the Christian worldview, there are three different unique sources that could cause visions (two for sure).
Its All In Your Head
The first explanation that is compatible with all worldviews is completely naturalistic- a mental state of affairs that causes the person to experience something as vivid as real life. This can be affirmed regardless of which worldview one adheres to. There is no guarantee that the content of such natural visions accurately reflects reality- especially from any non-theistic worldview- see Plantinga's evolutionary argument against naturalism. If the person who had the vision is a Christian and the person they are trying to convince is a naturalist, the naturalist is perfectly justified in rejecting the witness of the vision. The Christian would need to provide some supporting evidence.
In these states, the vision produced could support any claim about reality, or it could reject any claim about reality. Both possibilities make sense from this perspective. Since this explanation is compatible with all worldviews, anyone can explain a vision by appealing to the it- whether the content of the vision affirms or denies a specific worldview.
God Spoke In A Dream
The second possibility, that is only available in the theistic worldviews, is that God is the source of the visions. In this case, the content could be trusted. However, this is dangerous. If we wish to use only the vision as evidence, we have no way to test conflicting visions from people in the different theistic worldviews without assuming one of the worldviews is correct first. We would essentially be begging the question (using the truth of a worldview to prove that it is true). We are attempting to provide evidence for the truth of a worldview, so we can't assume it is true in our evidence.
We must test the claims of the two conflicting worldviews against reality, independent of the identify of the deity, to determine which one was possibly truly from the deity. However, the content of the vision can still match reality by coincidence- so, there is still a naturalistic explanation. This would be good if, say, a Muslim has a vision that accurately reflects reality, but is affirming Islam; the Christian does not have to necessarily grant that it was from Allah; there is the naturalistic explanation that may be appealed to, but this is not the only option.
Deceptive Spirits
The third possibility, that is also only available to the theistic worldviews, is that the source is a deceptive spirit (the other possibility in the case of the Muslim above). I tend to leave this third possibility in the air because I'm not sure that demons can cause visions. But just because there may only be the first two options, does not mean that a Christian would not have a consistent explanation for the Muslim receiving a confirmatory vision (or the other way around). I do want to include this in my analysis because many Christians do hold that it is possible.
Even with testing for the second and third possibilities, the naturalist could easily dismiss them by saying that the vision was a naturalistic phenomenon that simply confirmed what the adherent already believes. They could actually argue that if a person believed something already, then they are already in the state of mind to find confirmation. They could actually say that naturalism would expect such a confirming vision to take place. In this case, a vision could be used by the naturalist (or other worldview adherent) to argue against ANY other worldview's validity based on a vision of an adherent.
Because of that, I would never recommend that a Christian use their vision to try to convince anyone of the truth of Christianity. Visions are too subjective, and there is a viable naturalistic explanation which can be marshaled in any opposing worldview and used against the Christian's claim that their vision authenticates the truth of Christianity.
When Is It Safe To Appeal To A Vision?
There are only two ways that visions could legitimately be used as evidence for Christianity; however, they are very particular in the conditions that must be met:
The first is that the person who is in another worldview experiences a vision that tells them that their worldview is incorrect. However, this should not be immediately believed. The worldview that the vision claims to be correct needs to be tested with evidences from reality- again, it could just be a mental state of affairs, or it could be a deceptive spirit (from the perspective of Christianity, demons don't care which worldview a person accepts, just so long as its not the true one). If the worldview passes the test against reality, this vision could be used as supporting evidence, but it cannot stand on its own due to the other viable explanations.
The second one may be used as objective evidence AND may stand on its own. However, the vital condition that must be met is multiple, simultaneous experiencers. This is what the 500 and the disciples in 1 Corinthians 15 co-experienced. There is no naturalistic way that even two people can experience the same vision simultaneously. Pantheistic worldviews are the only ones that have a mechanism that could explain these naturalistically (the fact that all are one). Thus, as long as the person one is trying to convince is not a pantheist, visions may be used as objective evidence.
This would be the most powerful if one of the people is one who challenges Christianity (and even better if two of them are). The reason I say that is because if the person challenging Christianity does not experience the vision, they could dismiss the testimony of others by claiming collusion or that the vision never took place for even one of them. However, if they experience the vision, they have first hand experience that the vision took place. But they could still claim collusion if someone else (already a Christian) claims the same vision. If you had two unbelievers who experience the same vision at the same time, who neither were pantheists, they would have no mechanism for explaining away their mutual experience. The only way they could dismiss this objective evidence would be on purely emotional grounds.
Evidence or Subjective Confirmation?
I'm not one who thinks that actual visions from God are very common (at least, outside the Middle East). But when He does grant us that experience (even in the Middle East), I don't think that God gives them to us to use as evidence for someone else. He gives us visions in order to make the evidence more real and personal to us...to make Himself known to us in a deep personal way that objective, dry facts cannot do. The purpose is for the individual.
Part of My Testimony?
I know that many are tempted to use their vision (which they believe is authentically from God) as part of their testimony, but not necessarily as evidence for the other person to base belief on. However, this is dangerous. When we give our testimony, we are essentially telling the person that "What God did for me, He will do for you too." I see two issues with this. First, there is no guarantee that God will grant the other person a vision; He reaches different people's hearts in different ways. If an unbeliever is looking for a vision as confirmation of the truth of Christianity, yet it never comes, what then? Is Christianity false because God did not give the person the vision? That then has to be dealt with theologically (which it can, but is not likely to be accepted by the unbeliever). Second, we don't know for sure that the vision we had actually came from God. We could have been in the mental state that allowed us to have a confirmatory vision. If we give the person the impression that we had a vision and that God will give them one too, we are in the same boat as the person in my first issue.
Conclusion
Visions are powerful tools for convincing individuals who experience them, but they should not be used as part of our testimony or evidences that we provide. There are way too many liabilities. Now, I'm not saying that we can't communicate that we had a vision that we believed to be from God. We can. We just need to make it clear that our vision has independent support for its claims, and combined between the objective evidence and the subjective confirmation, we know Christianity to be true, we understand its significance to our lives, and have trusted Jesus Christ as a result.
See Also:
Argument From Religious Experience: An Interview With Dr. Andrew Corbett