The Paris attacks are transforming Europe’s migration crisis into a security debate, spurring calls for a clampdown on free movement across borders, and putting proponents of an open door for refugees on the defensive.

France’s firm belief that Islamic State militants planned the attacks—and the possibility that at least one assailant may have posed as a Syrian refugee—are fueling arguments over whether Europe is doing enough to protect itself from terrorists who might infiltrate the thousands of migrants arriving daily from...