Combine vinegar, sugar, salt, mustard seed, and basil in a large pot; bring to a boil. Reduce heat and simmer until sugar is dissolved, about 5 minutes. Stir green tomatoes, red bell peppers, red onion, and Granny Smith apples into vinegar mixture; boil vigorously, stirring occasionally, until vegetables have softened and relish flavors have blended, 15 to 20 minutes.

Sterilize the jars and lids in boiling water for at least 5 minutes. Pack relish into hot, sterilized jars, filling to within 1/4 inch of the top. Run a knife or 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 10 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.