From stopfoodwaste.org
Store cilantro and other fresh herbs just like cut flowers in a jar with 1-2 inches of water and covered with a plastic bag to maintain moisture.
For tender herbs like mint, dill, parsley, and cilantro, remove any wilted leaves, trim the stem ends, and place the bunch into a jar or glass with 1-2 inches of water. Try to ensure that the leaves stay above the rim of the container.
Loosely cover the herbs with a plastic bag and place them in the refrigerator for up to 2 weeks. Be sure to change the water every few days (or when it starts to look cloudy), and remove any wilted or slimy leaves when you notice them.
Tip: Don’t wash the herb leaves until you’re ready to use them as the extra water can rot the leaves.