In all developed countries and most developing countries, the central bank is charged with promoting stable money and a stable financial system. (This is true even though, in Australia and in many other countries, the task of supervising the institutions in the banking system has been allocated to a separate supervisory agency.) And these are extremely important goals.
To see just how important, it is worth reflecting on the economic and human cost incurred by many countries when central banks got policy wrong and contributed to the strong deflation of the thirties. Or reflecting on the economic and human cost incurred by those countries which experienced hyperinflation at some stage during the last century.