If a country wants to claim their place in the world as a beacon for freedom and democracy, said country must treat all peoples as worthy of those ideals. When "enemy combatants" are arrested for wanting to cause damage, even those people should be awarded constitutional protections as it is the way a country demonstrates their true belief and commitment to what they enshrine in their documents. Failure to do this adds to the hatred and disillusionment many in the world already feel toward the United States and its words compared to its actions.
Often times accused terrorists are profiled based on their family history or skin color. It would be entirely unfair and unethical to not grant these accused individuals the same constitutional due process that other citizens would have. This is the American way and it should remain true to the US Constitution.