There's some discussion of basement dehumidifying here https://greenwizards.com/node/813

Do you know if your hygrometer really gives a good measure? I've seen crazy difference between different hygrometers in the same space. Although, if you've got mold, that seems like a pretty good indicator that the surface is pretty darned wet.

You've got to improve your air circulation.

Sadly, unless you've got a prevailing wind and operable windows on all sides, that's hard to do without electricity.

Some suggestions to improve air flow:
ALL your shelving should be Closetmaid. That is, it should be vinyl coated wire shelving. This is easiest to do in closets and pantries. Closetmaid shelving is adjustable with a hacksaw.

Use vented louvered doors wherever you can. They don't work for bedrooms but for closets and pantries, they're great. We've also got a louvered door to the upstairs bathroom. Because of the way our house is built, that bathroom is only used by the person living in our bonus room attached to the bathroom. It might not work as well on a main bathroom, but if you need more air flow, you should think about it.

Bathrooms need venting, both a window and a ceiling vent.

Open windows all around, from top to bottom.
Make sure your attic is fully ventilated with ridge vents and soffit vents.
Basements need venting too, as do crawl spaces.

Yes, extensive venting can wreak havoc with heating and cooling so you want to consider if your vents need to be operable.

Sunshine can be helpful so more sun reaching deep into your interiors won't hurt either.

My electric dehumidifier can pull a gallon or more of water out of my basement in 24 hours, day after day. It's noisy and expensive, though it's the kind of job that would be fine running on intermittent (e.g., solar panel) power, because it wouldn't matter if it only ran during sunny hours, and missed a day or two. There are chemicals that absorb moisture, but these are only practical in sealed containers. If someone says that a bag of chemicals can dry your closet, or basement, ask yourself "where does the water go?" and "How do I tell if it's still working?"

If those solutions are not viable for you, I am in agreement with Lathechuck and Teresa up thread with the exception that I cannot quite get my mind around Teresa's comment regarding louvered doors and bedrooms.

Humidity and the problems of mold are a important issue for me, since I'm moving into the basement of the new place. I've not researched it as well as I should yet.

I'm focused more on initial issues of prevention of water entering the basement, and not yet lowering the humidity of the air in the basement. To do this, I first went around and patched all of the cracks I had in the cinder block walls. And I had quite a few too.

This involves chipping the cracks out until I had a 1/4" gap all the way thru the first wall of the brick and patching the crack with hydraulic cement. Working with it was actually pretty easy, like playing with PlayDow. I also gave the boundary between the wall and the floor a coat too. Here's a picture.

Next I covered all of the exterior walls with a 1" layer of polyiso foam board. Its the silver foil faced stuff with a vapor barrier on one side. It and the metal wall framing I'm putting in sit on a EPS foam subfloor. This is so that if I do get any leaks later, the water should (in theory) move to the bottom and then head towards the drain. The floor has a slight slope.

You can see the foil faced panels in this photo, as well as the subfloor. The pink panel under the grey kitchen cabinet.

After I get all the walls up then I'm going to be insulating the rim joists, which are the areas just above the basement wall and below the first floor floor, in between the basement ceiling rafter beams. And caulking all the windows for air gaps. I'll probably replace the old exterior door too eventually.

All of this is the cut any infiltration of exterior air as well as to provide a boundary layer between the cooler exterior walls and the floor and the warmer moist interior air. That's where you get much of your damage. Warm moist air hits cooler surfaces and the water gets deposited there, providing a place for mold to grow.

Common drywall has a big problem with that. It likes to retain moisture. I'm actually using cement board, which is pretty much moisture proof on the first 3 feet or so of wall, then moisture resistant drywall above that.

For comfort I'm still probably going with an electric dehumidifier, though the main living area, an office and a bedroom, will have a door to segregate them from the rest of the basement, as well as an additional 2" layer of foam insulation in both the walls and the ceiling.

In cold climates, you need a vapor barrier between the warm, humid indoor environment and insulation, so that the vapor barrier stays warm enough not to cause condensation (or freezing!) which would soak into the insulation. But in hot, humid climates, the vapor barrier has to be outside the insulation, so that condensation from the cold (air-conditioned) interior doesn't soak into the insulation. In my mid-Maryland temperate climate, the biggest energy expense is heating, so I assume that the vapor barrier (if there is one) is inside.

That's one MORE reason not to run the A/C down too far (in addition to household expense, and global environmental impact). We've got it set to 78F, and run a dehumidifier all summer. Sometimes, we'll knock it back to 76F for cooler sleeping (under a slow ceiling fan).

Given that we can trade-off temperature and humidity for equal comfort, I think it's best to go for warm and dry! Heat conduction (which the A/C counteracts) is proportional to the temperature difference, not the humidity difference.