This old post may be helpful: https://www.chronofhorse.com/forum/forum/discussion-forums/hunter-jumper/336975-outdoor-arena-watering-solutions-not-enough-water-pressure-for-sprinkler-system
It takes a tremendous amount of water to actually mitigate dust-- just making the top surface damp wouldn’t work because the drag right behind you will be exposing dry material underneath.
How much water, you ask? Arena length x width x depth of water x 7.5 = gallons. This isn’t perfect because of course the arena it’s not a impervious surface like a pool liner. So think about it in terms of how your arena “behaves” when it has rained. If you had a slightly rainy day where 1/4" of rain had fallen, the arena surface would probably be dust-free but not totally soaked, right?
So, if you have a 100x60ft arena, and wanted to put down the equivalent of 1/4" rainfall, that would require about 940 gallons of water. Even if you don’t need dust-free and just want to reduce the dust a bit, you’re still talking about hundreds of gallons to be worthwhile. Definitely need a tank rather than 5gal buckets
You could reduce the need for water by adding mag flakes or some other additive that binds moisture to the footing rather than just quickly evaporating.