I assume that the dishwasher drain connects to the sink (through a disposal?). Current practice is to put a high loop and a vent on the dishwasher drain hose to prevent water from backing into the dishwasher from normal operation of the sink. I understand that this wasn't the practice ~10 years ago.

Ponded water in the bottom of the dishwasher (maybe only in the pump so you really don't see it) can get stagnant between washer operation and cause an odor. You might want to check this.

The vent stack is connected to the sewer line, can you run water in it from the top to check if it's clear? (just a thought, I'm not a plumber and don't know if this is OK or not).

do all your drains function properly?
typically if your stack is plugged your drains will not function properly.
you could go up on the roof with a high power flashlight (like one those 500,000 candle power ones) and shine it down the vent. you would be able to see any obstructions

Also note most dishwashers have a pullout grate/screen over the central drain to catch larger food debris. These are often very flush-mounted and inconspicuous--look closely and feel around for the pull-tab. Pop it out and dump the rotting debris.

OTOH, auto-DW detergent is loaded with bleach, so it's rare for odors to originate in the DW... so ditto on checking that DW drain hose is routed up high before emptying into garbage disposal. This prevents GD from back-spewing waste into DW. Also never hurts to dump some bleach down the GD, followed 10-20 minutes later by strong detergent, followed a few minutes after that by a huge kettle of boiling water. GD should be ON during the boiling water dump.

As for the roof stack vents, if clogged, that will cause your P-traps to get sucked dry, allowing sewer gas to enter.
Might be hard to get a flashlight and your eyeball over the roof stack simultaneously. In that case, take the garden hose up on the roof and run water down the stack--if it backs up, it's clogged.

If clogged, leave the hose up there (turn spigot off), you'll need it later. Go buy a hose attachment called "Drain King" at the hardware store. There might be other brands, but basically it's a combination rubber bladder and nozzle that attaches to the garden hose.

When the water's off (or you're holding the hose kinked--hint, hint) the bladder is skinny and will drop down the stack. When the water flows, the bladder expands, locking the nozzle in the stack, so that the remaining water, shooting out the nozzle, can build up pressure and blast the clog free. Cheap and easy. Helps to have a helper on the ground, operating the spigot, and making sure nothing is backing up indoors. ;')

We don't really have garbage disposals up here. The drain hose from the dishwasher does a lazy S-shape under the sink and hooks up above the P of the sink.

Now I think the smell is coming from the furnace ducts!
The weather has been such that I haven't had to use the AC or the furnace. However, to test a theory, I've run the furnace fan by itself all day long for the past two days. The house smells with that musty old-pipe smell, which goes away when the fan is turned off.

That smell is NOT nearly as apparent in the basement, so I don't think the furnace is drawing the smell from down there. What is the smell of ducts that need cleaning?

Maybe the drain from the coils on your AC is clogged and there is water standing in there growing gunk. Can you look inside by the coils to see if there is water?