Was Keynes wrong?
No. The first respondent said "yes," and in the interest of intellectual discourse, I will acknowledge that (I'm new to the site, maybe that answer is still here in another response?). But that person left no reasoning. The typical argument against Keynesianism, if it is not driven by ideological zealotry, is that deficit spending is inflationary and will eventually meet its day of reckoning, when it will only have exacerbated the problem. Thus many argue that economies must suffer "short-term" pain in the interest of long-term stabilization through the market.
There are a few problems with this. One is that people literally die in the short run, so there is much human suffering that Keynesianism can prevent. Another is that contractions that are allowed to go unabated cost the economy things that it needs to recover and grow. Take education for example. In many African and Southeast Asian countries, when their economies have fallen on hard times, governments have implemented "austerity," or contractionary policies. Often on the chopping block is free public education. When families on the edge of subsistence are forced to pay for schooling, often they respond by pulling their children, especially girls, out of school. But apart from gender equality issues, losing out on schooling hurts the whole economy. Education is one of the things that improves a country's ability to produce goods and services efficiently (technology is another). A less educated workforce means less potential for recovery over the long run too.
There are many other reasons government spending helps to alleviate recessions and depressions. It can save enterprises from bankruptcy. While saving one or two is unimportant, if most enterprises in an economy were to fail, a catastrophe would ensue (there are numerous historical examples of this). Unemployment skyrockets. People out of work reduce spending, so that more businesses fail, more people lose jobs, etc. Economic contraction is a vicious cycle that government spending can counteract.
As for the costs 1) It is a documented fact that the government spending multiplier is bigger than the tax (tax cutting) multiplier. Raise taxes on the wealthy. They are spending a much smaller than average portion of their income in ways that generate economic growth. Often they are speculating in comodities or stocks, driving up prices of things like food, or creating dangerous bubbles in financial markets. Other times they are hoarding massive reserves of currency, preventing businesses from getting access to capital they need to grow. 2) The growth that government spending can stimulate will enable the economy to get bigger and stronger than if no intervention was made, so that even without raising tax RATES, government gets more tax revenues, simply because people make more money. In this way, government deficits can be paid.