Line a wire-mesh strainer with 3 layers of cheesecloth or 1 (12-cup) coffee filter. Place strainer over a bowl. Spoon yogurt into strainer. Cover and chill 24 hours. Remove yogurt, discarding strained liquid.
Preheat grill to 400° to 450° (high) heat. Combine strained yogurt, basil, next 3 ingredients, 1/2 tsp. salt, and 1/4 tsp. pepper. Cover and chill until ready to serve.
Toss together okra, olive oil, and remaining 1 tsp. salt and 1 tsp. pepper in a large bowl.
Grill okra, covered with grill lid, over 400° to 450° (high) heat 2 to 3 minutes on each side or until tender. Cool 5 minutes.
Transfer okra to a serving dish, and serve with dipping sauce. Garnish, if desired.