First, make sure you explain why the information is secret. If the community understands why certain information cannot be disclosed, they're a lot more willing to accept decisions based on that information, even if they're not allowed access to it.
Second, make sure there is oversight. If one moderator's decision is questioned, bring in other moderators to look at the facts and weigh in.
Last, make sure there's an appeals process. On Stack Exchange sites users can appeal to Community Managers via email if they're not satisfied with moderator decisions.
One other point that I would stress is to give community members the benefit of the doubt if there is any. You want to establish a reputation of making fair decisions, especially when the information you make those decisions based on is kept secret. You can't do that without the trust of your community.