For fiat currency, the government's role is crucial. This is not the only viable means of monetary policy, however. A free market could possibly produce a viable currency (probably based on a gold standard).
Intervention is justified when there is market failure. Best example would be when the costs are somehow externalised. That is, when you privately transact, you end up costing other people, or benefiting them. To fix this, a government needs to internalise these costs somehow, either by installing taxes or incentives to achieve the socially desirable equilibrium (as opposed to the equilibrium that would be achieved in a private market).
In reality though, with lack of ideal conditions to produce fully efficient markets, there are a whole bunch of minor things that could justify intervention. But it would be short-sighted to ignore the reality that government intervention is never done ideally - political agenda overrides economic rationalism and often the costs far outweigh the benefits - an economic solution is rarely put forward cleanly.