Removing a ballpoint pen stain from silk can be done in three easy steps, with supplies you likely already have in your home. It's best to treat the stain as soon as possible, because dried-in ink is much harder to remove. It's also important to keep in mind that silk is a delicate material, and before using any product you should test it on a small, inconspicuous spot before applying to the entire garment.
Step 1 - Prep Work
Place the soiled silk on a clean, dry white towel. Place it stain side up on a firm, flat surface.
Step 2 - Clean
Spray the stain with an alcohol-based hairspray. At first the liquid may appear to spread the ink, but this will dissipate with cleaning. Blot the stain with a dry towel using constant pressure. Use a different and clean part of the towel with each dab to prevent spreading the stain. Continue to spray and dab until the ink disappears.
Step 3 - Follow Up
When you have removed the ink, launder as normal.