Much will depend on what kind of caulk was used.

If the caulk is silicone, it needs to be removed, area cleaned good with bleach, etc. and then new caulk.

If it is a porous type grout, it is possible bleach will kill the mold.

A fan blowing in the bathroom aimed at shower/bath tub after use will reduce growth of mold. Mold loves warmth and moisture.

Try getting some Tilex mold-killing bathroom cleaner spray and use that on the tub area. Then use a fan to prevent re-occurrance. Be careful of the fumes and don't mix Tulex with other cleaners. I open the window when I clean my bathroom.

It will end up saving you time in the long run to simply re-caulk the area; make sure you clean the area with bleach first prior to re-caulking.

Here is a link that might be useful: College Girl Cleaning Service