Is an EZ cloner dishwasher safe?

Anyway you need to sterilize it between uses. A mild bleach solution, washed off well, will kill any mold spores.

The bown rot is what is called an 'embolism', which means you have trapped an air bubble in the vascluar tissue of the stem while taking the cutting. This will eventually rot the stem.

In the future, take your cuttings under water with a new razor that has been wiped off with bleach, and get them into the cloning gel immediately.