You have to work at establishing that before you get into a crisis situation. If the public goes into it with suspicion or cynicism toward leadership, there is nothing much that leadership can do to gain that trust in a crisis. He and others agreed that trust is essential for crisis leadership. There is a great deal more cynicism. I think the answer to that is very simple.
Leadership needs to tell the truth to people.
There may be issues of timing, of when you tell people things. But it is a crime against democracy to affirmatively mislead people about critical events, unless withholding information is necessary to prevent further bloodshed or violence or something like that, and even that should be viewed as temporary. People can't make informed political decisions if they're not informed. Trust has to be a belief this person can do what he's supposed to do, substantively. It's an evaluation of capability as much as it is of believability.