How to keep shower caulk clean
I have used three things in the past:

Toilet cleaning bleach gel every other month or so.
Spray bleach cleaner occasionally.
Bleach/baking soda paste, with diluted bleach spray every few hours to keep moist.  Paint on with a cheap paintbrush as the bleach will destroy the bristles.

Once you have given up hope, use the third option and cover the paste with saran wrap to keep it from drying out.  Overnight should save it.