Managing mildew/mold???

The best advice I can give is keep everything vented. Windows open (install screens), louvre doors for closets (leave some open) have a ceiling fan or a pedestal fan running on low speed.

You can treat leather things with a water resistant spray to minimize mildew/mold.

Put shoes on a rack so there's air flow possible. We hang belts and don't roll them. My leather laptop bag hangs on a hook in my office, my wife's leather purses are on a shelf and not in a closet.

Then, check your stuff on a regular basis and when needed clean it and leave it out of the closet for some time.

If you must keep stuff in boxes, use moth balls. But for the most I would agree with Gary the best preventive action will be keeping good ventilation. Specially during this rainy season.

Is there any tricks to getting rid of the mildewy smell after the fact? I see tips for prevention, but what about after it's already a problem? The cupboards in the house I'm renting smell gross (mildewy) despite my cleaning with sanitizing products.

Tried Clorox?

A mix of 10% clorox should do the trick.

Bleach. Plenty of fresh air---a fan to circulate the air.  A product called Damp Rid.   Light--natural or artificial.

People hang their clothes to dry in an outside cloth line which dries it with the air and the sun. Then they are moved into the house. Better than a clothing drier. Bleach is your friend for surfaces and plenty of air. Make sure line is not metal or it will erust. Using plastic line instead and clean it with bleach from time to time.

For smell, re wash and hang to dry, maybe use fabreeze detergent

look at microban, it keeps detering mold for a long time.  popular product in baton rouge after august floods.   bleach is very good but once it dries it is done.

We are having that problem, with our new bamboo shades, from Home Depot