While refugees are indeed people and do have rights, citizens of the US have rights too. A country is obligated to protect its citizens, but it isn't obligated to protect the citizens of other countries. Look at the violence and terror in places like Germany and France. Many (not all, to be fair, but many) of the terror attacks of the last few years have been perpetrated by so called refugees. We can't allow potential threats into the country and put our citizens at risk.
While after we fix our immigration system, we'll be morally required to take in refugees, right now we just can't take the risk that refugees potentially pose.
We are well within our rights to refuse entry to anyone from anywhere for any reason. If we continue to take in anyone who spouts a pitiful sob story, we will soon be destroyed from the inside. We must reduce the number of threats we allow into this country, and not allowing people from known terrorist hotbeds is a good start. It doesn't go far enough, but it is a good start.
Syrian refugees should not be prohibited from entering the country. There are security problems with them entering the country, but security issues will still be a problem whether or not they enter. There are other solutions instead of prohibiting them. The country can increase our security and screening process. We can also take them in, in smaller numbers making the transition easier for the country and the refugees.
The only thing refugees are looking for is equanimity. The only reason the refugees fled from the country because their country is no longer a safe place for them to continue life. Their existence is threatened. There's no reason for them to ruin the country whose already accepted them. We are human, we are belong to humanity.