Unless the battery box is a stand alone compartment intended only for the batteries, and it has its own cover, then covers are usually required by survey/insurance.
Because this battery box does not contain anything else and nothing is stored in here the wood cover meets the criteria. The terminals should still be protected to prevent accidental shorting. These flooded batteries are in plastic acid proof boxes to contain the acid in the event of a spill or cracked case.
Any time you have flooded lead acid or gel batteries they need to be contained in a acid proof battery box large enough to contain the acid in the battery.
This one would also meet the criteria for no cover other than the wood cover sitting there. They are also in acid proof battery boxes.
Also don't forget that anything connected to the + post of the house battery bank should be fused within 7" of the positive post.