What is the punishment for losing an access badge?

Yes, by all means report the loss. Here's a little story from my wilder days. In 1999, while my battalion was doing OPFOR for the 45th Bde, who were preparing to deploy to what used to be Yugoslavia for peacekeeping missions, I noticed that an E5 had left his notebook on the checkout counter at the mini PX. Being the kind and generous fellow I am, I snatched up the notebook and took it straight to my S2. For the next week, we were able to monitor 45th Brigade communications, locate and attack the Brigade TOC three times, create havoc with their logistics, and start a small witch hunt by posing as one of their company commanders claiming to have found a weapon, M16A2, belonging to one of the sergeant's men. Shortly after which, the Brigade Commander called the sergeant's boss and ordered him to report to the Bde TOC with the sergeant an all of his troops for a 15-6 investigation. Using the information in that one, little notebook, we were able to sew disaster, and defeat the Bde's attack on our TOC, because we knew his plan. One little notebook. Report the loss, SPC Rogers.