I agree with the posters that state your sink has to drain completely
of water to avoid the mildew buildup to begin with. While I cannot
offer a solution to the sink drainage situation, I can give you 2
interim solutions to mildew - if you have a kitchen window above the
sink - open the shades/curtains to let the direct sunlight in. Or, if
you don't mind the temporary smell, put some vinegar in a spray bottle
and spritz the bottom of the sink a few times. The acid in vinegar
kills existing mold & mildew (might need a toothbrush to clean the
crevices) -- and prevents future growth (if applied regularly) without
using any caustic chemicals around food/dishes.

peter wrote:
Would a new sink be more immune to mildew? What about metal sink? Are there
kitchen sinks made with a gentle slope at the bottom such that water would
roll off but dishes can stay put?