Curling leaves on basil could be a couple of different problems. Often it’s because the plants have a sucking insect at work on the underside of the leaves – these insects pierce the leaf cells and this causes the leaves to curl as they grow. Sucking insects that attack basil include white fly (tiny, TINY white insects around the plants) and aphids for the most part. Aphids gather on the new growth and are visible.
Another cause of curling foliage could be a fungal disease such as powdery mildew or botrytis mold, or a water mold such as downy mildew. Powdery mildew and botrytis look like light grey powder and downy mildew looks like dirty, grey powder on the underside of the leaves.
Finally, fertilizer burn can cause distortion of the leaves if a synthetic fertilizer is applied in too great a concentration or if it’s given to a dry plant. Never fertilize a thirsty plant.
The bottom line is that basil is an herb that grows best outside, in hot weather. Indoors it’s more prone to all of the problems listed above. Basil can be planted outdoors when the night time temperatures are reliably above 50. So once you see that it will no longer go below 50 at night, you can plant basil outdoors and it will be less prone to problems.