In the event you get your running shoes wet from the rain or going through a puddle or even if your feet have sweat a lot, here's what you should do in order to dry your shoes.
1. Remove the insoles and set them to the side to dry.
2. Loosen the laces and pull the tongue out and forward a bit to open up the shoe so air can circulate in the shoe.
3. Stuff some crumpled up newspaper into the shoe. This will absorb some of the moisture, especially if your shoes got soaked.
Check the shoes after a few hours. If they are dry, remove the newspaper. If they are still wet or damp, remove the newspaper and replace it with new dry crumpled newspaper. Do not use a heat source such as a hair dryer or heat lamp to try and speed up the process. This can potentially damage your shoes, as well as being a safety hazard.
Allowing your shoes to dry out completely will help prevent that unpleasant odor. The part of the shoe that will hold most of the smell is the insole/sock liner. You can remove the insoles from the shoes, wash them with some soap and water and let them air dry to get rid of the smell.
If that does not work, you may want to consider trying out a shoe deodorizer. Check out our list of the 10 best shoe deodorizers to find one that works best for you.
In addition to the above, you might want to consider having a second pair of running shoes, so that you can rotate them. This will allow each pair of shoes to fully dry in between uses.