The early Cold War period was the golden age of American capitalism because it coincided with the rise of the so-called World War II generation, the cohort of then young men (and a few women) born between 1915-1925, that fought and won World War II.
A grateful American public passed the so-called GI Bill that paid these ex-soldiers' college tuitions. In turn, these were unusually mature students who had already received an "education" in the trenches; George Bush Sr. completed a four year program at Yale in 2 and a half years. Shaped by wartime experiences working together, and working with the latest technology, these young men then went out and reshaped American business. A group of ten "whiz kids" who had gone to the U.S. Defense Department, then Harvard Business School together, entered Ford Motor Company as a team and effectively took it over.
These men entered their midlife managerial years during the early stages of the Cold War, and one of them, Ronald Reagan, won the Cold War with the "Star Wars" program late in his managerial career.
Expanding on Graviton's answer, the Marshall Plan played a large role in the USA's economic development.
Europe was devastated from the war and no longer able to export goods in large quantities, thus also decreasing its ability to import, especially from the USA, which the Allies had greatly relied on for WW2 weaponry and supplies (food, textiles, ammunition, raw materials...). Losing such a big market would decrease the USA's national income, so it pumped money into Western Europe to revitalize it, aiming for long term gains in the future.
If it was a Golden Age for American capitalists, it's probably because they got a lot of countries like Greece (which was facing Communist-Royalist Civil wars) to support them through Harry Truman's Doctrine, the Marshall Plan and NATO, all of which came into operation before the mid 1950s, which I would regard as the early Cold War period. And by getting all this support, they actually stopped the spread of Communism in those countries. This is the sort of thing that capitalists then would call 'victory'. Apart from this, you can consider factors like the first man to land on the Moon being from a capitalist country.
First of all, I would challenge your statement that wars are bad for the economy. On the contrary, I believe history would inidcate that wars are actually very good for the economy. During periods of war there is higher employment, more manufacturing and exporting (especially of war materials), and usually more innovation.
After World War II, a lot of jobs were created because of the need for rebuilding. While almost all of this rebuilding happened in Europe, there were several American countries involved with it, resulting in boosts to the American economy.
Furthermore, even though the "Cold War" was not an actual war, a lot of the same factors were in play. The US was still manufacturing weapons at a high level as a preventative measure, but that still meant that a lot of people were employed. The increase in defense spending resulted in an increase in disposable income which in turn resulted in more spending, which meant the wealth was being spread among multiple sectors.
I'd say that your premise is incorrect. Wars are not inherently bad for an economy. Look at World War Two, it pulled the US economy out of the Great Depression before the US even entered the war, through the Lend-Lease program. War creates massive demand for goods because things tend to be destroyed, so they have to be replaced. That demand has to be supplied, which requires a build up of industry.
After WWII was over, the industries which had switched over to war time production, like the auto industry, used the factories they had built during the war to produce cars at a comparable rate. A similar trend occurred in other sectors of the economy.
The GI bill also helped the development of suburbs and cities. The veterans who came back were more educated through it as well.
In the immediate post war era, the US was the first large economy of its kind to get the kind of things we take for granted now. I'm talking TVs, basic appliances, etc. Home ownership and automobile ownership. Why exactly the US was the first to get that explosive growth in all these areas is not something I am qualified to answer, but it's important to note that perceptions of America's 1950s "Golden Age" is in part due to the simple fact that America then had a huge advantage over other nations just because it was getting things first. We no longer idealize TVs, refrigerators, mass car ownership, etc. as huge economic achievements because many other places have them. European countries experienced explosive economic growth too, but it took a little longer with them.
Also to add to the other responses above, it should be remembered that economic rebounds after a long economic slump are often that much more powerful depending on the economic slump they are following. When recovery does come, the perception of recovery fuels more spending, more investment, more optimism and hence a stronger recovery. And of course, economic slumps breed immense opportunities for growth. The People's Republic of China provides an extreme example of this; after several decades of Maoism and really fairly terrible economic growth, the PRC economy went through the roof in the 1980s, and that was due, in part, to the simple fact that the PRC was starting from somewhere close to zero economically speaking, when compared to the rest of the world, so it had nowhere to go but up. Though much less extreme, the pattern was also evident in America in the post war era.
Overall, it can be argued that America's post war "flourishing" was really a relative perception that we have looking back on it, based on how we were doing compared to other nations now.
Other posters have ignored perhaps the most important factor which led to the US economy flourishing during the early Cold War (post-WWII) era:
The United States emerged from WWII with the only undamaged industrial base in the world
All of the other major pre-war players - Britain, Germany, Japan, etc - had all been smashed to bits by the war. The United States had been the industrial engine that kept the Allies going, by producing much of the machines and weapons needed to fight the war. Afterwards, America was the only source for a lot of consumer goods - and even then it took the US economy several years to re-tool to produce consumer goods, resulting in a pretty severe post-war recession. But once those factories were putting out consumer goods they had a ready market for whatever they made - the rest of the world. In the post-war era goods didn't have to be particularly innovative or even well-made (witness the insanity of "planned obsolescence") - they just had to be there, and they'd sell. For 25 years after WWII it was basically impossible for an executive to screw up a big American corporation as the rest of the world would buy whatever was produced because there was effectively no alternative.