In cryptography, the objective is to keep the messages secret. The enemy's objective is to break the code and determine what the messages mean. The more clues you give as to how your messages are encrypted, the easier it is for the codebreaker to succeed. However, if the people who are encrypting their messages discover that their code is broken, this destroys the value of breaking the code, as the message senders will simply switch to another code. This is why, for example, why we didn't use all of the intelligence that we gained from breaking the Enigma cipher - because acting on information that could only have been gained by breaking the cipher would have let the Germans know it was broken - destroying our ability to gain further information through the broken Enigma code.
In the instant case, in order to prosecute Padilla in a civilian court, a discovery phase is mandatory. In the discovery phase, the prosecution must reveal to the defense the information they have that proves the case. During wartime this means that in order to have such a trial, we must necessarily reveal what we know about Padilla's terrorism links - and in the process, giving vital clues to the enemy as to how we know it.
If the discovery process reveals to the enemy the methods we are able to use successfully to obtain intelligence, this will cripple our ability to gain further intelligence by those channels and put at mortal risk our agents in the enemy camp. The enemy will stop using the methods and means of communication that are vulnerable to interception. The end result of this is dead US soldiers and spies and a diminished ability to pursue the war (which is the end goal of the alleged "civil liberties" crowd).
So in the end, it is a question of what is more valuable - giving Padilla a civil trial, or wartime intelligence. I have no problem with giving Padilla a civil trial after the war is over (though it may be decades). Thus I conclude that the military justice system, being able to keep secret information secret, and also being more appropriate to the subject - Padilla is accused of an act of war against the US, after all - is a more appropriate venue, given the crimes alleged of Padilla.