Some hot days I will close the door to the basement, take a panel off of the furnace so it is drawing air from the basement, and then turn the furnace fan on. I notice the difference right away.
Understandably, you may not care to do this because of odor or fears of radon.
Team door closed. Cold air falls and hot air rises. The best you would be doing is letting the cool air from your upstairs fall downstairs. Even if you could get the cool air from your basement upstairs, it would likely be too humid.
More importantly as others have said you need to address whatever is getting your basement wet. Water is your #1 house problem, from the roof down to the basement. If water is getting anywhere outside your plumbing system you have problems. If your basement is getting wet many things you store down there will get mildewed/molded and ruined, especially permeable surfaces like wood, leather, paper, etc. Glass and plastic should generally be ok.
If it’s just a little bit of water a fan blowing to circulate the air can help.
I seem to remember reading some where that there was a proposal in some city to use cool air from the city’s subway tunnels to cool buildings located near the line, back before air conditioning was common, but I can’t find a cite for it now. I think it may have been the never completed Cincinnati Subway. Although in that case I think the idea was that passing subway trains would push air up through a ventilation shaft into the buildings.
Mice can get through very small holes. Unless his basement door is sealed like an exterior door, keeping it closed is not going to stop any mice who want to come up and have a look around.
Nah, basement door has a gap at the bottom of it that lets me fit 2 fingers.
The door will not stop mice, so that’s irrelevant to O.P.
Mice do not use doors. Every winter some mice get into our home. I set traps in the basement and catch a few there. I also set traps in our upstairs bathrooms because they get in there via the water lines.
To exhaust the warm air out of the house I put a fan in the upstairs window of a north facing bedroom. It helps tremendously.
This works even better if you can open a window at ground level on the north side to suck in cooler air, especially if you make a ‘shade house’ on that side of the building.
Leaving the basement door open allows the Things that live down there to get into the rest of the house, especially at night.
This assumes they don’t already know how to open the door. 
I also set traps in our upstairs bathrooms because they get in there via the water lines.
You might be able to discourage them somewhat with steel wool stuffed tightly in any gaps around the pipes.
There’s a logic to that, but just make sure there’s also a clear path for air to get back down to the basement from the rest of the house, rather than sucking it through the basement rim joists, or down the water heater flue. It’s also important to consider that sucking hot air out of the rooms is just as important as blowing cool air into them. Short-circuiting the return air can lead to weird balancing issues and unexpected behavior.
As for radon, it’s an issue in basements because it both seeps in from the ground below and it’s heavier than air, so it tends to settle down there just like cold air does. I wonder if mixing around the basement air with the rest of the house would disperse the radon enough that it goes below the threshold of concern. Plus it makes the basement air now subject to the infiltration from the rest of the house, so not only is its concentration reduced since it’s mixed with air from the rest of the house, but it also has more chance to escape to outside.
My furnaces have had a “summer switch” which circulates the air from the basement throughout the rest of the house. It lowers the temp by a couple of degrees and makes a small difference in air movement. My new furnace has the fan switch incorporated into the thermostat unit so I don’t have to trudge downstairs to switch it on and off anymore.
The previous owner of my house said they’d leave the panel to the attic removed in hot weather. They said it created a “chimney effect”, drawing hotter air up into the attic and pulling the cooler air up from the lower level. I have no idea if it worked or not. I later found rodents in the attic and was unwilling to leave the panel off anymore.
If you have access to the base section of the cold air return duct you could put an intake vent there. One that can be opened or closed. Ideally you also put a blocker above it. That can be opened or closed. Close the blocker, open the intake vent. Then turn on the furnace fan. This will suck air in from close to the basement floor. Then force it up into the house. Unfortunately all your heat vents are likely in the floor. So the cool air will not rise too much, and be sucked back down to the basement at low height. But it might get things cooler. Leave the basement door open. If you don’t install a blocker above the vent, then block off the cold air return vents at the floor level.
Can someone please explain what that means? (As a reminder, this house has no air conditioning, just a window unit. If that’s ^^ related to central AC it’s gonna be meaningless.)
The furnace has a hot-air-goes-out side and a cool-air-comes-back-before-getting-heated-up-again side. If you put a vent in on the cool air side ductwork (low down, I believe) in the basement, the furnace fan will suck in more cool basement air and blow it back upstairs (while the furnace heating element is off). I had one added to my previous furnace ductwork for very little money. It worked okay before the vent was added, but it worked better after it was added. Someone else might be able to explain it better.
I just moved into a two-floor apartment and am currently in camp “leave door open” because there is no door to my (mostly finished) basement But also, I don’t have air conditioning, so I put one of those big industrial-style floor fans at the base of the stairs to blow the cooler air up. It’s extremely effective! My hallway is now filled with a noticeable cool breeze which circulates across the entire floor. Note though, there are two sets of stairs here, so the air can head back downstairs via the other stairway.
The fan looks something like this, and it’s pretty big (20+"):
To be honest, I’m not sure how much it’s actually cooling the upstairs, or if the breeze just makes it feel cooler… Also, the basement becomes much warmer by day 3 of a heat wave, but it’s still cooler than the ground floor.
Gah, I didn’t know about possible radon! I just bought a test.
Q for those knowledgeable about dehumifiers – I just bought one of those, because it’s pretty humid here. I ran it for a bit, but it just seemed to make the air hotter due to its exhaust being so warm. Any tips? Is it better in the long run to just run it anyway?
I have a radon meter as the radon level is higher then we would like on the charcoal test, but below the limits(4.0 IIRC). It was around 2.2 with doing nothing. When we ran the a/c fan 24/7 it would drop to about 1.2, this appeared to be because the lower level only had a exhaust vent, not a intake, so by using that it would pressurize the lower level, and ‘fight back’ against radon entering from the ground. We then got a sub-slab depressurization system which drops the level to about 0.5 no matter what. But it was amazing to see that lower (basement) pressurization actually worked, though I never saw anyone using that. I was actually considering installing a duct/fan to take air from the top of the house and pump it to the lower level to both pressurize to keep radon lower (without the need to run the a/c fan all the time) and also help mix the air, putting the warmer and dryer air at the lower level. With the radon issue solved I still think it would be a good idea to help stabilize the house in temperature and humidity to undertake such a project, but at the same time the house is recently constructed and I don’t want to tear it up too much like I would with a older home.