Add strawberries, basil, lime juice, and 1 cup of water to a blender. Blend until smooth.

Pour the mixture through a fine-mesh strainer into a large pitcher, pressing on the solids to extract as much liquid as possible. Discard solids.

Stir in the remaining 3 cups of cold water and agave nectar or honey to taste. Adjust sweetness if needed.

Refrigerate for at least 30 minutes to chill.

Serve over ice, garnished with lime and strawberry slices if desired.