A badger hair brush is a solid investment in your daily shaving ritual, and as such, deserves at least a minimum amount of care to ensure that it lasts as long as it can.
For starters, thoroughly rinse your brush and razor after each use. Shake all excess water out of your badger hair brush so it can easily dry. Dry the badger hair brush by hanging it upside down to prevent moisture from seeping into the base of the brush. If you do not have a brush stand, you should seriously consider one. They can be very inexpensive if made of plastic, or very expensive if made of nickel, silver, or gold.
Never store a shaving brush in a closed cupboard where mildew may form. Replace your blade as soon as it begins to uncomfortably tug and pull on your beard (about every 3 to 5 shaves - a heavy beard growth will wear out a blade quicker).