When it comes to the distrust of the state of banks’ balance sheets, transparency about the true nature of the assets on the balance sheet will clearly help. But the transparency and the disclosures clearly need to be credible to reduce the information asymmetries.
Stress testing of the balance sheet by a credible independent party such as the prudential regulator (APRA in the case of Australia) can give depositors, and other investors, confidence that their money will be safe. These sorts of actions work to take the solvency risk off the table. For depositors, this is reinforced by the presence of deposit guarantees.