Stain removal is all about speed and technique. Here's the proven method for tackling ink stains effectively.
Put paper towels underneath the stain to absorb the ink as it releases. The ink needs somewhere to go other than the fabric.
Saturate the ink stain with rubbing alcohol. The alcohol dissolves ballpoint ink. Blot with a clean white cloth -- don't rub.
Blot the stain with a fresh section of the cloth each time. Move the paper towels underneath as they absorb ink. Repeat until no more ink transfers.
Wash with Power Wash Laundry Detergent on the warmest setting safe for the fabric. Check the stain before drying.