Congress has broad and extensive power to do, without question, what is necessary to raise a military force in order to defend the security of the United States.
The information on the card simplifies the system by making easier for selective service applicants to be in contact with the selective service boards. The ability to communicate allows the government to call up individuals for military service, which is of vital interest to Congress.
Draft cards can be illegally changed or modified. Destroying the cards entirely makes it almost impossible to catch the people who attempt to dodge the draft by altering them. This significantly inhibits the interest of Congress in creating a military very quickly to respond to threats.
When individuals possess, and can therefore see, the information on draft cards, they are reminded if they need to update that information if it changes.
The information on the draft card still serves a compelling interest to the government as showing the card to an official upon request quickly helps ensure its bearer is following his selective service requirements.
The local board needs to be able to keep track of the whereabouts of potential draftees in order to be able to call them up quickly if the need arises.