Alex asks this question. I am not so enamored of the pre- and post-Fed macroeconomic comparisons. Presumably the Fed advocate also favors deposit insurance and active exercise of the lender of last resort function. Furthermore most Fed advocates today do not want a gold standard, in part because it ties the hands of the Fed in a crisis. I view 1929-1932 as a better illustration of the workings of "a world without a Fed" than "a world with a Fed," even though of course we had a Fed then. If you take the relevant division to be "before and after WWII," the Fed looks pretty good.
It's also the case that the 19th century had much less interconnected leverage than today's world (not only because of the absence of a Fed) and that agricultural productivity was a much bigger determinant of overall volatility.
I see the historical ledger as follows:
1. The Fed made 1979-1982, and the 1970s in general, much worse than they had to be, and for no good reason. Count that as a strike against the Fed.
2. The Fed made the recent crisis much better than it otherwise would have been. Without a Fed, we would have experienced something akin to a Great Depression, including a frozen payments system.
2b. If you wish to give the recent crisis an anti-Fed spin, you can argue that if previous bailouts had not occurred, we might have avoided the high levels of leverage circa 2006 and thus avoided such a major crash. We would have had one or two big recessions earlier on, however. More to the point, I believe that Congress would have done earlier bailouts; after all, that is what just happened in Ireland and TARP was Congress in any case. What is the point of going that route? People think of the gold standard as fetters on the Fed, but I think of the Fed as an excuse for Congress to step back. Consult the wisdom of Garett Jones.
3. A "real Fed" would have made 1929-1932 much better than it was. In my view this #3 more than cancels out #2b. It is unrealistic to ask for a perfect or even a very good Fed, but it is not unrealistic or utopian to advocate "a Fed better than the 1929 Fed" and indeed we've had that ever since 1937.
4. In the 1950s, 1960s, 1980s, and 1990s, I see the Fed as bringing improvements, of unknown magnitude.
5. Historically central banks have been essential in helping nations fight major wars. The world's preeminent military power simply will have a Fed, for the same reason that it has lots of nuclear weapons.
6. More generally, both Fed and Treasury are usually, in relative terms, voices of economic reason within government, even if they're not everything you wish them to be. It is arguably counterproductive to lower their status. Currently, the relevant alternative is a totally politicized Fed, not no Fed at all; see #5.
Addendum: Bryan Caplan offers relevant comment.