Place strawberries in a food processor; cover and process until pureed. Transfer to a large glass bowl; add vinegar and lemon zest. Place 1/4 cup basil in a small bowl. , With a mortar or a wooden spoon, crush some of the basil until aromas are released. Repeat with remaining basil; stir into strawberry mixture. Cover and let stand in a cool, dark place for up to 3 days, stirring once daily., Line a strainer with 4 layers of cheesecloth or 1 coffee filter and place over a large saucepan. Strain vinegar into pan (do not press out solids). Discard solids. , Heat vinegar to 180° over medium heat. Carefully ladle hot mixture into hot half-pint jars, leaving 1/4-in. headspace. Wipe rims. Center lids on jars; screw on bands until fingertip tight., Place jars into canner with simmering water, ensuring that they are completely covered with water. Bring to a boil; process for 10 minutes. Remove jars and cool.