In a bowl, mix the salt and dried basil; spread on a small plate. Moisten the rim of a glass with the lime wedge; turn the glass in the salt to coat.
In a food processor, pulse the peaches and sugar until smooth.
In a cocktail shaker, muddle the fresh basil. Fill the shaker halfway with ice; add the lime juice, tequila, triple sec and 1 1/2 oz. peach puree. (Leftover puree will keep, refrigerated, up to 3 days.) Cover; shake vigorously until cold, about 15 seconds. Strain into the salt-rimmed glass.