It also makes him Commander in Chief during times of war.
"The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States..."
One of the things that Commanders do in wartime is try to listen in on communications between enemy combatants to try to determine their strategy and stop them. It's called Signals Intelligence.
Understood but, IMHO, not really relevant here. One, Congress has not declared war. Two, the commander-in-chief clause generally does not grant domestic law enforcement powers to the federal executive short of scenarios involving civil war, invasion, or insurrection; domestic law enforcement, constitutionally speaking, is a state matter.
Bottom line: wage war without quarter on terrorists abroad, by definition they are ineligible for Geneva Conventions protection. Domestically, however, different rules apply. If the authorities have probable cause that a crime has been committed or suspect a conspiracy is afoot, they can obtain a warrant per Amendment IV. Claiming constitutional authority to declare anyone, even American citizens on American soil, as enemy combatants subject to indefinite detention without trial, as the current administration has done, is risible.
Moreover, the government can always secure the border and largely prevent the entry of enemy combatants to begin with. Government claims that we are at “war” would have more credibility if officials seriously addressed the border issue - securing the perimeter is also sound military practice.