Dust falls, so start at the top.
The Closet Shelf Nobody Looks At
How to clean a closet from the top shelf down to the floor, and check it for damp.

When did you last look at the top shelf of your closet? Dust settles there, on the rod, and on the floor, and it stays out of sight until something comes out smelling stale.
Cleaning a closet takes a few plain steps, in order from the top down. If that's too much today, the top shelf alone is a good start.
Take everything out

Take everything out of the closet and set it on the bed.
An empty closet lets you reach every shelf, corner, and edge of the floor.
Dust from the top down

Dust the top shelf, the rod, and the walls, in that order, with a dry cloth or a vacuum brush attachment. Use a stable step stool, not a chair, for the top shelf.
Dust drops onto whatever is below it, so working from the top down means each part gets cleaned once. A chair can tip, and a step stool is made for reaching high.
Wipe the shelves and rod

Wipe the shelves and rod with a cloth dampened with water and a drop of mild dish soap, then dry them with a second cloth.
Damp wood or metal in a closed closet stays damp and can grow mildew, so drying right away matters.
Vacuum the floor and corners

Vacuum the floor, using the crevice tool along the baseboard and in the corners.
Lint gathers along the edges, and the crevice tool is narrow enough to reach it.
Check for damp

Look at the walls and corners for dark spots and notice any musty smell. If you find either, wipe the spot with a damp cloth, dry it, and leave the closet door open so air can move through.
Mildew grows where air doesn't move and moisture lingers, and it can spread to clothes stored beside it. If it comes back after you wipe and dry the spot, the wall may be damp from a leak, and a repair person can check.
Put back what you want to keep

Put back what you want to keep, leaving a little space between items.
Air moving around clothes and boxes helps the closet stay fresh.
Short on energy?
Just dust the top shelf, from a stable step stool. That's enough for today.
The larger lesson
A closet stays fresh because of what surrounds the clothes as much as the clothes themselves: clean surfaces, dry walls, and room for air.

Clean from the top, and the floor only needs one pass.
