That much rot is pretty conclusive evidence that the deck boards aren't cedar or redwood and that they're not treated. Definitely seal them. There exist copper-based wood preservatives that you could use to kill the mold on the underside and keep it from coming back. They can be price competitive with deck sealers, so you might instead use such a product on the underside if you intend to use sealer underneath. You might not like the color of these wood preservatives, so on top you would probably not want to use them. Definitely take care not to get blots of stain on the exposed surface.
The tightness of your board edges looks like a problem. This configuration won't allow water to drain through from above, and it won't allow water vapor to vent from below. The International Residential Code requires vented crawlspaces to have 1 ft² of ventilation for every 150 ft² of floor space, so I would space the boards to roughly achieve this level of venting between the boards. Remember that the boards swell with seasonal humidity, so provide a bit of extra space if you do this work during a low humidity season.
Ground water from below could be a problem source. Grading the surrounding area can mitigate this some. A french drain is an alternative option. Gutters on the roof edge above the deck can also move water away from the deck. I suspect that the roof edge and poor surface drainage caused your really badly damaged boards, but I'm uncomfortable asserting it without more images.
While you have the deck boards removed, take a look at the framing below. If this framing is untreated, then the bearing points on top of those pavers is a problem. Especially if you decide against any ground water mitigation, consider spraying wood preservative around the wood in contact with the pavers.
