When this happens it is always unfortunate because the common result is that good people leave the organization disillusioned with the leadership and discouraged with the lack of concern for the staff. Usually, by the time the issue is dealt with, some of the best people are gone.
Why does this happen? First, boards rightly assume that staff issues are the purview of the senior leader so they don't get involved. At one level this is correct. Boards should not be giving direction to staff apart from their senior leader. But at another level this is flawed thinking. If the senior leader were taking the organization in directions that were disadvantageous to the organization, the board would step in. Where there are serious morale issues, those issues are a threat to the organization - if the staff involved are good staff that the organization wants to keep. Healthy boards never ignore threats to the organization.
So how does a board keep a pulse on staff in a church or ministry non-profit. Informally, conversations with staff where the board member is not giving direction but simply listening to how the ministry is doing can be helpful.
More formally, the board can ask for reports on any trends regarding resignations from the organization. Such reports are consistent with policy governance and certainly can give a board a heads up if there seem to be common issues.
Third, there are software programs that can measure engagement of staff and general satisfaction with their work. Such programs can be a great help to senior leadership and boards have every right to see the monthly results as well.
If board members believe that there is an issue that needs to be addressed with staff morale it ought to be a topic in executive session and then raised with the senior leader. And there needs to be a way to verify what is true and if there is a plan to deal with morale issues, whether it is successful. While boards need to give their senior leader wide latitude in assessing and solving morale problems they are also ultimately responsible for the health of the organization so cannot ignore the issue.
I am even more concerned when boards seem totally unaware of serious issues within the staff. What this tells me is that the board has inadequate policies or procedures in place to monitor the health of its most important asset: the staff. If it matters to monitor the financial situation of an organization it matters just as much to monitor the satisfaction of the staff. Both are active indicators of the organization's health.