After much experimentation, I find its better not to wash cilantro until you are ready to use it. Before refrigerating, chop off the heavy steams and remove the leaves that are going bad. Pat dry the cilantro. Wrap the cilantro in the paper towel and store it in a paper bag. I like using brown paper lunch bags.
I have been able to store cilantro for almost 10 to 12 days.