In a 6-quart stockpot or dutch oven, heat the blackberries and apple juice over medium-high. Stir in the pectin, and bring to a rapid boil. Add the basil, and up to 1 cup of honey, depending on your tastes. Return to a boil for 2 minutes.
Remove from heat, fill jars, and process in a hot-water bath for 10-15 minutes.