If you are not keen on using bleach, there are a number of other solutions you can consider for getting rid of mold.
One method that can work well is to combine water and hydrogen peroxide in equal parts. You can use a spray bottle to dispense the hydrogen peroxide. Wear rubber gloves as needed if you are cleaning mold this way.
You will need to wait for a little bit while the hydrogen peroxide goes to work and remove mold from shower caulking, and then you can rinse it off.
Another option to remove mold from a shower caulk is to combine water and baking soda. You can try adding just a little baking soda to the water, then spray, wait, and wipe it down. Alternately, you can adjust the ration to make a baking soda paste and use a scrub brush or toothbrush.
Yet another shower caulking removal solution is white vinegar. You do not need to dilute the vinegar with water. Unlike a water solution, you can pour this right into a spray bottle.
Spray the vinegar to remove mold from the shower caulking, and then wait. After the vinegar has had some time to work, you can rinse it off to remove mold spores from the acrylic caulk, bathroom door, or shower tiles.
If you want to enhance your vinegar, you can add a bit of tea tree oil to your spray bottle. Spray it onto the shower caulking and wait before rinsing off the silicone caulking. With all of these mold removal methods, your shower caulk should be just fine.