Combine the vinegar and sugar in a small saucepan over medium heat. Bring to a simmer, stirring often to dissolve the sugar. Remove from the heat and allow to cool.
Put the cantaloupe and honeydew in a bowl and toss with the basil and mint. Drizzle with the sweetened vinegar and toss lightly. Pile the melon on a platter and sprinkle with the flake salt.