In a 1- to 2-quart pan over high heat, bring sugar and 1 cup water to a rolling boil. Nest pan in a bowl of ice and stir often until cold, 8 to 10 minutes.
In a food processor or blender, purée tomatoes, a portion at a time. Rub through a fine strainer into a bowl; discard any residue.
Combine syrup, tomato purée, and basil. Freeze in an ice cream maker according to manufacturer's directions or until dasher is hard to turn.
Serve sorbet softly frozen, or freeze airtight until firm, at least 3 hours.