Liability of owners cleaning dumpster enclosure?

I expect you mean cleaning is more then just picking up trash that failed to make it into the bin.

Regardless if the Association is aware of it, it's possible that Association may be held liable if something happens.
If the Association does know about it, it's more likely.
If the Association provides cleaning equipment, then they most certainly risk liability if something happens.

With all of that said, if the Associations desires to have the owners clean the enclosure, then they can minimize the liability by having the owners sign waivers (said waiver should be drafted by an attorney). In this situation, it likely wouldn't be an issue providing equipment (brooms, dustpans, gloves, etc.). I would be hesident on providing cleaning solutions or bleach as that would increase the risk if the Association also didn't provide proper training on the use and cautions of the solutions.

I will also add, if the issue is simply trash that didn't make it into the bin (or what animals may have brought out of the bin) I don't see a problem with the Board putting a broom and dustpan in the area. Don't advertise it's there and don't publish in a newsletter or email chain asking that owners use the tools or clean the area.

The Board can always publish that residents are reminded to make sure that their trash is properly disposed of and placed inside the bin.

Check your insurance policy! And check your state's Workers Comp requirements. A waiver or Hold Harmless agreement will not, by itself, protect the association from an uninsured injury or other claim. If you want volunteers doing physical labor, you need to have both.

I will refrain from my usual rant about Why Using Volunteers May Not Be As Smart As You Think It Is...

****Their own*** dumpster enclosure . Do all the individual units /homes/ lots have a dedicated dumpster , enclosed by some fence ? I don’t know why someone cleaning their own trash enclosure is a problem . Or is there a dumpster for six units/ owners that get serviced by waste disposal contractor , who charges not only to pick up the waste and also sweep and clean the enclosed area . Or does a hoa maintence person make the rounds and pick up . Are these enclosure in a public area that attracts non residents with trash to throw away ?