In a large bowl, whisk together the beer, flour, cornstarch and 1/2 teaspoon salt; let rest for 10 minutes. In a small bowl, whisk together the mayonnaise, tarragon, mustard and garlic; season with salt and pepper.
In a large pot, heat the oil until it registers 375 degrees on a deep-fry thermometer or a drop of batter sizzles immediately and rises to the surface. Whisk the batter again, add the green beans and toss to coat. Working in 4 batches and using tongs, gently lower the beans into the oil. Fry, turning halfway through, until golden, about 3 minutes. Let drain on paper towels and season with salt. Serve with the aioli.