I think that the most powerful element of the closing of the Western frontier was the reality that "it was running out." The fact that the reality of Manifest Destiny had met its natural end helped to bring to light the idea that there needed to be other areas where American expansion could be present. I think that it was the closing of the American frontier that helped to build the idea of imperialism and setting national sights abroad. In this light, the closing of the frontier had tremendous impacts for millions of people who did not live in the United States. The closing of the frontier also signaled the reality that capitalism and wealth will eventually endure its own sense of closing or "stagnation" where expansion will be impossible. I sense that this is the significance of such an action.
Another very important influence of the frontier on American history and on its present status as a nation and empire was the magnetic effect the vast quantities of unsettled, untapped land had for immigrants from other nations. The fact that much of the west's farmland was free or exceptionally cheap brought millions of people from Ireland, Germany, Italy, Poland, Russia and other countries to the United States. The draw of this farmland permanently changed our population and our culture, and their labors in addition to their numbers made it possible for the country to become an empire.
So the closure of this frontier, which was inevitable at some point, put the brakes on long term immigration (though it still continued) or at least did so for those who were mainly motivated by the prospect of free land. It also marked the end of gold fever, and the oil boom, and the promise of instant and fantastic riches that the resources of the frontier offered in the minds of businessmen and tycoons.
The closure of the Western frontier was historically significant for the United States because 'the close of the frontier' gave way to what would become known as 'American Expansionism'. I usually refer to this period as the United States' 'muscle flexing period'. The close of the frontier was the motivator behind the powerful force of the industrial capitalists. Whether considered the 'Captains of Industry' or 'Robber-baron the industrial capitalists' of the late 19th century saw not the close of the United States but the financial potential beyond California. The close of the Western frontier was in essence the beginning of the United States' journey as a world power.
To me, this is a question of opinion since you cannot really measure the effect of this closure.
I would think that you could point to a couple of impacts of this closure.
First, you could say that this represented the first time that America was not a frontier nation. That is to say that it was the first time that it was not engaged in expansion. This deprived people of a place to go when their current situation was not to their liking. A person with few prospects in the east could no longer try to find his fortune on the frontier. You could argue that this led to the creation of an urban working class and to the reduction in America's pioneer spirit.
Second, you can argue that the closing of the frontier helped lead to overseas imperialism. Perhaps the closing of the frontier pushed America to try to expand outside of the bounds of North America.