You don't actually need to seal the MDF so it doesn't absorb moisture, any more than studwork in walls, the interior components of cabinetry or the framework in covered chairs need to be sealed. Regardless of whether it's solid wood or a type of manmade board the material can be left to absorb and release moisture naturally with any changes in humidity. In any case the whole concept of "sealing" is commonly misunderstood, with only a few exceptions the coatings applied to wood don't truly seal the surface off from water vapour*.
However, it looks like the natural colour of the MDF won't suit the completed panels so I think you need to paint them.
Normally when someone is looking to paint MDF the recommendations are about how to get the surface looking good since it'll be on show (e.g. filling/sealing the edges, then primer and finally paint, with quite a bit of sanding thrown in there). But here it seems like you really only need to change the colour so you could go straight to paint. Matt black seems like it would be the ideal colour.
*This is why even a fully varnished tabletop will still expand and contract through the seasons as humidity goes up and down.