While the humanitarian situation in Syria is of critical importance, the risk of losing ancient historical sites is also an issue that has to be addressed. There are six Syrian cities, including Damascus and Aleppo, listed on the UNESCO (the cultural branch of the United Nations) World Heritage List; in a letter to President Obama several cultural preservationist groups, including the United States Committee of the Blue Shield, asked the president to keep those sites in mind in the event of a military attack. 

The letter reminded the president of his obligation to protect Syria's cultural heritage, as laid out during the 1954 Hague Convention. The groups also asked the president to issue executive orders asking federal agencies to "encourage and enter into agreements with any allies and any rebel forces with which it coordinates, including the Free Syrian Army, to ensure protection of Syria’s cultural heritage" and to ensure that looted artifacts aren't easily distributed.

So far the damage has been dishearteningly severe. In June of this year, the Associated Press reported that UNESCO listed all six of Syria's World Heritage sites on a danger list, noting that several mosques and citadels have been damaged and an "illicit trade in Syria's rich archaeological heritage has flourished" over the last two years.

UNESCO expert Francesco Bandarin told the Agence France-Presse that artifacts were already popping up in Beirut. "On some markets, there is a real influx," he said. The image below shows a damaged minaret (left side) and what it looked like before. 

From the Associated Press:

The minaret of a famed 12th century Sunni mosque in the northern Syrian city of Aleppo was destroyed Wednesday, April 24, 2013, leaving the once-soaring stone tower a pile of rubble and twisted metal scattered in the tiled courtyard. President Bashar Assad's regime and anti-government activists traded blame for the attack against the Umayyad mosque, which occurred in the heart Aleppo's walled Old City, a UNESCO World Heritage site.

The world has learned, to an extent, from similar looting and military actions in Iraq, Afghanistan and Libya. The Iraq National Museum was able to save several artifacts by hiding key works and blocking off the museum's entrances with cement bricks. Maamoun Abdulkarim, Syria's antiquities chief, told the Associated Press that Syria's government also managed to stash away the country's most valuable relics.

As for the antiques still out in the open, there's not much America can do. Based on current laws, the U.S. will attempt to prevent stolen artifacts from entering the country. But in the event of a strike (assuming recent diplomatic developments fail) the U.S. has a history of firing at or near historic landmarks