Sounds like you may have grey mold, which is caused by a fungus in the soil. The fungus is most active in cool, humid weather and is spread by splashing water, people, or infected fruit.
Avoid overhead watering by using a soaker hose or drip irrigation. Mulch with straw, pine needles, or other material to keep fruit off the ground. Allow enough space between plants for good air circulation and rapid drying after rain.
There are chemical treatments for mold, which I'm hesitant to recommend for fruits.