Stakeholders influence corporate policy, values, and general direction. They advise the board of directors and they sometimes have the ability to vote in members of the board as well.
Stakeholders of public companies also influence the price of the stock based on their buy-hold-sell behaviors. When stakeholders buy and hold on to their shares, prices tended to stabilize and move upward. When stakeholders buy and sell their shares quickly, prices tend to destabilize and move upward.
In health care organizations, many stakeholders express ideological and political concerns which might move the company in a certain direction. For example, if a large group of stakeholders -- controlling a large number of shares -- put pressure on the organization to provide indigent or low-cost health care, it is likely that the company will move in that direction.