In contrast to the classical viewpoint, Keynesian economists start with an assumption that an active monetary policy can make real and permanent changes to output levels if it is enacted at times when the economy is under-performing i.e., when there is spare-capacity in the economy, a money supply boost can help the economy to expand.
Empirical evidence on the matter has shown fairly conclusively that the neutrality of money does not exist in the short-run, and that real economic variables are indeed influenced by monetary policy shifts.
However, in the long-term it appears that the classical economists are correct. Active monetary policies do not make any permanent changes to the economy, only prices are permanently affected, whilst output and employment levels will return to their natural rate. The process by which this occurs is explained on my page about the:
Dynamic Models of the Neutrality of Money
Whilst the static models are useful for highlighting general concepts, they are limited in their ability to paint a big enough picture of real-world changes caused by monetary policy shifts.
The dynamic models of the neutrality of money go further than highlighting general concepts, and attempt to look at the wider implications of money-supply changes on a constantly adapting economy.
Since the central assertion of the static NAIRU model is that a permanently higher inflation rate will be the only long-term result of an expansionary monetary policy, we need to take this analysis to the next step and investigate the long-term effects of higher inflation.
I have answered this question in some detail on my page about:
That page is instructive on the ravages of inflation, but if you don't have time to read it then it suffices to say that the effects of higher inflation are highly undesirable and unfair. The worst impact of high inflation is leveled on the poor, because there is a significant re-distributive impact of rising prices that takes money from the poor in favor of the rich.
Clearly this is a serious issue, and it demonstrates a serious limitation of the static models' sole focus on aggregate measures whilst failing to account for the changing composition of national income that results from monetary expansion.
We can deduce from the preceding sections that it is only in the very loose sense proposed by the static models that the neutrality of money can be said to exist. Even then, empirical evidence has convincingly shown that there is no neutrality of money in the short-run, and that the neo-classical economists have failed to demonstrate any adherence to the rational expectations theory to which they align.
Money is not neutral in its effects on real people, not in the short-run, and not even in the long-run once we go beyond static analysis and look at the more comprehensive dynamic models.
Nevertheless, the question that remains relates to optimal policy choices.
If an active monetary policy boost/contraction can help to smooth out the boom-bust business cycle then the benefits of using it should outweigh the costs. Unfortunately, the evidence of the monetary authorities' success in this area is overwhelmingly absent. The current state of the national debt and the mounting budget deficit is testimony to that.
The weight of evidence suggests that active monetary policy has actually had a destabilizing effect on economic output, and when the re-distributive effects of the resulting inflation due to the non-neutrality of money are added to this destabilization, the costs borne by our society, particularly the poorest members of it, have been severe to say the least.