Mix mulberries and 1 tablespoon water in a large pot; bring to a boil. Reduce heat, cover pot, and simmer, stirring frequently, until berries are softened, 15 to 20 minutes. Mash berries using a potato masher.
Combine 2 cups water, sugar, strawberry-flavored gelatin, and pectin in a bowl; stir until gelatin is dissolved. Stir gelatin mixture into mashed mulberries; bring mixture to a boil.
Sterilize the jars and lids in boiling water for at least 5 minutes. Pack the boiling mulberry mixture into the hot, sterilized jars, filling the jars to within 1/4 inch of the top. Run a knife or a thin spatula around the insides of the jars after they have been filled to remove any air bubbles. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with lids, and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars into the boiling water using a holder. Leave a 2-inch space between the jars. Pour in more boiling water if necessary to bring the water level to at least 1 inch above the tops of the jars. Bring the water to a rolling boil, cover the pot, and process for about 15 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool. Once cool, press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.
