Wait, are you saying the 12v+ at the plug is hot all the time? LOL
Seriously, the first place to start is in your owner's manual, which you can find online
HERE.
Look in the fuse panel in the engine bay, should be by the battery. Pull off the plastic cover and look at the fuses. This is from page 10-34 in the owner's manual:
Location 68 (circled in red) is the fuse for 12v+ power at the 7 pin plug. If there is a fuse there pull it and see if it's blown, or if it's a dummy. In my series of Suburban they left the fuse location empty and supplied the fuse in a baggies in the glovebox along with the pigtail for the brake controller. I have heard that later models were shipped with a dummy fuse installed. So you have to actually pull the fuse and see if it's a dummy or blown.
99% that's your problem.