Fresh berries are expensive and can go bad very quickly. So how do you keep them fresh for as long as possible? First off, Eater reports that you should only wash as many berries as you plan on eating. (Water exacerbates the rate at which mold will grow on fruit.)
As for the berries you're not eating, store them in the fridge in a container that can breathe. You should also place a dry paper towel at the bottom of the container to soak up any excess moisture. If you've tried this and still ended up with moldy berries after just a day or two, rinse your berries in a mixture of one cup white vinegar and four cups of water, then drain and dry them thoroughly.
Store them as you would unwashed berries, on top of a dry paper towel in an open container in the fridge--the idea is that vinegar kills mold spores on berries, which, in turn, prevents the fuzzy growth.
Source: Extra Crispy