Water movement shouldn't be needed for storage. I'd add some circulation, though, just before use, if I was doing a large water change. There might be some nutrients in the saltwater that could drive a bit of bacterial growth.
I have a power head and air pump in my brute container (with lid on). I mix continually for 2-3 days checking salinity till I get it where I want. Then I put the powerhead/air on a timer running it for an hour three times a day.