Stinky shoes plus non-circulating air -- such as in a closet -- equal awful, obnoxious indoor odors. While you can't keep the closet door closed forever or leave your shoes permanently outdoors, you can reduce or even eliminate the stink with natural odor-eating substances such as baking soda or even dry coffee grounds.
Identifying the Culprits
The first step to solving the odor issue is to identify which shoes are contributing to the offense. Most likely, some pairs of shoes don't smell bad at all. Go through the closet and set the non-noxious shoes aside. As for the stinky shoes, set them in a separate pile and also look at where they've been stored. The items they're stored on or in, such as hanging shoe organizers, may smell as well. Remove all the items that typically touch the stinky shoes so those materials can also be de-funked.
Removing the Stink From the Shoes
Now that you've figured out what's foul and what's not, give the stinky shoes a treatment with natural deodorizers. Sprinkle 1 or 2 tablespoons of baking soda inside each stinky shoe and boot, shaking it around so the baking soda coats the insole area. Set the shoes outdoors on a sunny day -- or at least a dry, non-humid day -- for several hours. Dump out the baking soda and stuff the shoes and boots with wads of newspaper before putting them away; the paper helps absorb remaining odors. Treat other stinky areas, such has hanging shoe organizers, with a baking soda sprinkle, or spritz them with vinegar if the material won't be damaged by liquid. Wash items such as stinky storage bins in water with a squirt or two of mild dish soap and several tablespoons of white vinegar. Allow all the stinky items to air out outdoors for several hours.
Cleaning the Closet
If the closet's floor is carpeted, sprinkle it with baking soda as well, vacuuming up the powder after 30 minutes or so. Place bowls of vinegar in the closet, and close the closet door for another 30 minutes; then open it and dump the vinegar down the drain. Vinegar absorbs odors much the way baking soda does, and vinegar's own scent dissipates quickly once the vinegar is removed from the area. To keep the closet fresh, hang mesh bags of activated charcoal -- buy it at a pet supply store -- from hooks in the closet, or use bowls of dry coffee grounds or baking soda, replacing these odor-eaters when they no longer smell fresh. Air out the closet on a regular basis to keep it smelling its best -- poor air circulation and damp conditions could lead to stale or even musty odors, beside just the funk from the shoes. Run a dehumidifier or air conditioner in the room during humid weather while the closet door is open to help keep excess moisture and potential odors out of the closet.
Shoe Moisture Maintenance
Some types of shoes get stinky when they're wet; both leather and synthetic athletic shoes may reek after a run in the rain, for instance. Allow those saturated shoes to dry outside on a mat on the porch or balcony, or in the garage or another place with adequate ventilation, rather than the closet. In a closet, the shoes will take extra long to dry while they -- and the closet -- get extra funky with foul odors. Never put the shoes in the closet when they're wet or smelling sour; instead wait until the shoes dry, and either sprinkle the insides with baking soda or stuff newspaper inside them to help prevent strong odors.