A responsible nation takes care for the integrity of its borders. Just as a responsible homeowner takes care that windows and doors can be closed securely. Not that doors have to be locked all the time if the neighborhood is safe. But just letting complete strangers come and go without knowing who they are or what they're doing is asking for trouble. Even if a cop or city worker comes to your door, you check to make sure that they are wearing the right uniform and carrying the right credentials, and you don't let them wander around without knowing where they are. That's not racism or paranoia, that's basic responsibility.
This metaphor extends to the Syrian refugee situation. Someone comes to your door claiming to be afraid for their life. Might be a neighbor you know, might be a stranger. Might be an abused spouse, or depending on the neighborhood, someone running from a gang fight. You want to help, but will helping put your family in danger? Do you keep the door shut, or do you let them in?
If you do let this person in, you still have to be careful. It's a bad world, and while you want to be a good host, and you don't want to treat them with automatic suspicion, especially if they are really running from trouble, you are taking a risk. It could be a scam. You have to exercise caution.
Obama has ridiculed state governors for not wanting to take in refugees. He says that they are afraid of "widows and orphans". If it were just widows and orphans, there would be no issue here. The opposition to Obama's refugee resettlement programs comes from a lack of trust in the federal government's ability to act with that basic caution I'm talking about. This administration has a history of action based on snap judgment and ideology.