It really matters how the ring is constructed, what kind of drainage there is, how much water is expected, and how many horses there are that are going to use it.
It also depends on the labor available to drag the arena.
Well engineered surfaces are rideable in and through the rain, especially lightly. Well engineered surfaces are also extremely expensive. In places like southern California, rings tend not to be heavily engineered because most of the time you can get away with just grading and compacting and throwing some sand over it. In addition, these rings may be in places that get very heavy traffic, not just one or two horses a day.
I have ridden in rings in California that were never closed for rain and also ones that were closed constantly.
If a heavy rain is expected, it’s nice to ‘seal’ the ring by leveling the footing perfectly to encourage even penetration and runoff. The challenge is, maybe your people work until 5 and you don’t know when the rain will start. So you seal the ring at 4 even though it may not rain until the next morning. And then you lock the gate to keep horses out.
Whatever is happening, it’s not worth trashing the footing in the ring if conditions are muddy. It can be a good day to go out on the trail when that’s an option, or just take meandering walks around the barn roads.