This is part of a set of indoor furniture made to hold vinyl records. It's cut from walnut veneer 3/4" plywood. I think the inside is poplar?

I live with two boys under 5, so it needs to be somewhat demon proof. My question is 2 fold...

1. What should I finish it with? Oil? Poly?

2. (probably the most important) HOW in the world can I do this? I have access to a sprayer, but I don't think that will really get all the little holes enough. I would be willing to hand brush/cloth each hole, but I'm worried it would inevitably get really sloppy and drippy.

Any ideas?