I have had barn board on the lower half of my basement walls for 30 years. They were never painted, stained or finished and look great today. However, the room has a bit of a musty smell. Would it help to seal this wood today?
How should I seal it?
At the end of last summer I too discovered that my basement had a musty smell. However before painting to mask the smell you must locate the source of the smell. For me it was some moisture that got behind the sheetrock on one of the concrete walls, and a bunch of mold grew as a result. So I tore out the sheetrock, used a mixture of water and bleach to clean up the black mold that was causing the musty smell. I sealed the crack where the moisture was getting in. Then I primed the area with Zinsser's cover stain and applied two coats of finish.
Unless the wood has started to rot, or termites got to it somehow and that is causing the musty smell? If the boards have been on for 30 years with no problems I would leave them alone. I suspect your problem is moisture seeping in from somewhere else. Perhaps the wall behind them? Foundation showing any signs of moisture? Gutters are not clogged and causing water to seep slowly into the basement? Sump pump float need adjustment?
Has it been exceptionally wet outside where you are?
I would try a dehumidifier (or extra or larger one) to see what happens.