Upon seeing the picture of a Syrian/Kurdi refugee child dead on a beach, the world, especially those people more affected by social media, became concerned and demanded action. Soon enough, developed western counties began posting offers of various amounts of refugees they would take in their territories. The United States was slow and reluctant in the bidding for accepting Syrian refugees, but it was not without reason.
The Syrian civil war is a proxy war. No matter if the Cold War officially ended, US and Russia are still playing the geopolitics game. And the Syrian civil war, despite the agency of Syrian rebels and activists, and that of the government, is part of a global conflict between US, who backs the rebels, and Russia, who supports the Syrian government, for political influence in the Middle East. One may argue that the rebels are not fighting for US nor the government is fighting for Russia. That's true, but that was the situation of any proxy war. The Korean war was fought mainly between Koreans, yet there is quite an agreement that it was a proxy war between Russian Communism and American Capitalism.
Whereas the Syrian war is a proxy war with support of the United States for some of the sides, Syrian people affected by the war are very likely to blame the United States for the civil war, or its development. More than once, United States has been accused of supporting ISIS, and there is no need to getting into the trouble of verifying such a claim. An accusation is enough to enforce peoples opinion, just because US has not remained neutral towards the war. US has been found to meddling into other countries affairs before, therefore its government and intelligence agencies don't have a good reputation in many people's minds. People blaming the US for the destruction of their own country is too much of a risk for the US to take, while the American population still has a fresh memory of the Tsarnaev brothers (who blamed the US for the Palestine refugee crisis, not even their own country).
There will a large backslash from important sectors of the American citizenry. Some people think that there are too many immigrants in the country, some are concerned with national identity, and some feel unfairness, as too many Middle Eastern countries are not taking any refugees. Israel is taking none, Saudi Arabia is taking none, Dubai is taking none, so why does the West have to take all the burden? These people think that the Middle East people should take care of their own people, and they are right. On the one hand, Middle Easterners, when not accepting Syrians as refugees, are rejecting their own folk, as if Syrians where not one of them, on the other hand, western countries, when receiving Syrian refugees are expanding their notion of one's own folk to all of humanity, because nobody cares about somebody else's problems. Why can't the Middle Easterners do that?
Besides all the trouble of taking in Syrians, the United States have a better option to save face concerning the refugee crisis. Most of countries pledging to receive Syrian refugees are doing so by publicizing a number of refugees to take. As soon as you start count anything it becomes fungible. If the offer of one country was to accept parents of their residents, families with children, wounded, single mothers, orphans, or even if one of them conditioned the influx of people to job offers from interested business, or looked for people to take in according to their occupation, or something like that, it would stating a quality for the people to get in. As they pledge assistance in numbers, they are somewhat treating people as fungible. The United States could just one up everyone and open their borders, by example, to South Sudan refugees. There is a lot more war refugees in Congo-Kinshasa than any western country dare to receive, but, as war in Africa get so much less media coverage, only the people affected seem to care. And it does not seem so easy to blame the US for the Sudanese civil war, so it would be more safe. I am not being cynic or utilitarian, that would be suggesting to the US government to take in Venezuelan refugees, as that would demoralize Maduro, who is an enemy of them. But there are those drug wars in Mexico, so why not receive some Mexican refugees while the world takes care of Syrians? Every life matters anyway.
Paradoxically, the US government would get more positive media coverage by showing concern about people who are not getting any media coverage. And there are so many refugees in the world that they can choose.