I short-term and longer-term store with them down. In my driveway between trips, I keep the jacks down since we're typically in and out of it several times per week....sometimes doing "stuff" and sometimes just sitting in ti wishing we were somewhere else. I don't like to run the slide out without the stabilizers down, and when I'm out there just hanging out, I like to have the slide out.
As for storage during winter, I put the jacks down because I wanted to take the weight off the tires to keep the gravel from pressing into the treads too deeply while sitting in the same place for 3-4 months. Although I store indoors, the floor in the facility is gravel.