Here's what I'd do:
- remove all old caulk,
- let shower dry for several days,
- clean with a bleach like substitute and let sit overnight,
- seal the grout with melted beeswax or caulk
- and that should do it.
For more information:
Read "How can we safely clean up mold? Our 1820 home was closed up for 15 mos, and developed a white mold problem." a Q&A answered by Harold Remlinger.