To some extent, you could say Prospero does not actually trust others once he's on the island (other than Miranda). He uses his magic to control Ariel and Caliban and he uses it to control the others once he gets them on the island. So he is not trusting them to do things on their own.
However, I think you can say that he trusts the shipwrecked men or at least some of them. For example, he wants Ferdinand to marry Miranda. This means he trusts that Ferdinand is a good person. He also trusts the others enough to get rid of his magic, believing that they will now treat him fairly.