Use An Airtight Container Or Bag. Try to eliminate as much air as possible by using an airtight container or a zip-top bag with any excess air pressed out. I recommend a container that fits the sugar as snugly as possible, so there isn't a lot of extra air trapped in the container with it.
Try A Terra Cotta Disc. Once your sugar is soft, you can use a terra cotta disc to keep the sugar soft. (Soak it for 20 minutes, blot excess water, then add it to the sugar for 3-6 months)
Avoid Light And Heat. Don't store your brown sugar near the stove to avoid heating the sugar. Ideally, you'll store brown sugar in a dark, cool area (like a pantry, in a cupboard, or in opaque containers).
Use It! Lastly, try to use open containers of brown sugar in a timely way. Keep an eye on expiration dates, and use your oldest brown sugar first.