1/4 cup freshly squeeze lemon juice (more to taste)
Sterilize lids and jars for 10 minutes in a large pot of boiling water. They can stay in the hot water after sterilization until you're ready to use them. Put a few small, clean plates in the freezer to chill.
Put the blackberries in a large, heavy-bottomed saucepan and use a potato masher or wooden spoon to lightly crush them.
In a food processor fitted with a metal blade, blend the basil leaves, sugar and orange juice until smooth. Stir the mixture into the blackberries, add water and bring to a simmer over a medium heat, stirring occasionally until the mixture is thick, about 25 minutes. Stir in the lemon juice, taste and add more if you'd like. Remove jam from heat.
Using a clean spoon, drop a bit of jam onto one of the chilled plates and angle the plate down a bit to see if the jam runs. If it runs quickly, the jam hasn't been cooked long enough yet. Put jam back on the heat and try this test again in a few minutes. If the jam moves on the plate only a small amount, it's ready.
Set a large stockpot filled with enough water to just cover the jars on high heat. While waiting for it to boil, skim off any foam that's settled on the top of the jam. Ladle the hot jam into the jars, leaving 1/4-inch headspace. Remove any air bubbles and adjust the headspace, if necessary. Wipe the rims with a clean cloth and seal with the lids.
Process the jars for 10 minutes in a boiling-water bath. Using tongs or a jar lifter, remove from the water and set on a towel to cool and seal. To test the seal, run your finger over the lid after 30 minutes. You should feel a slight depression in the center of the lid. The sealed jars can be stored in a cool, dark place for up to one year. If a jar fails to seal, put it in the refrigerator and use within one month.