Replacing moldy sealant

There are a few spots I've neglected to replace old brittle sealant (around the kitchen sink) or never sealed at all (new faucet I installed 8 months ago). Now there's black gunk accruing under and around the sealant.

When I go to replace the sealant, how do I kill the mold so it's safe to seal over again?

Thanks for any advice you can offer.