I have started to store berries in the refrigerator only because I have noticed that there is a species of parasitic worm that feeds on ginseng berry pulp. Similar to the worms that feed on husks of black walnuts. So in the fridge they go.
First, I would be careful with taking the berries out of their immediate location with plans to bring them back in the future to replant. Id check with your local rules for doing such.
Second, the red berry pulp serves as a protective moisture barrier (natures refrigerator) to the seed inside, this is why I never pop the berry when I replant it. Planting the seed yourself compared to the plant just dropping it significantly increases the regrowth rate. If you are going to store the seed, keep them moist (not soaked) in a refrigerator. You could even take the seed out of the berry at this point and just store the seeds. There are a number of options but keeping them moist and cool are the key.