How do doctors correct congenital cataracts?

Depends. If the cataract is visually significant and treatment is indicated, they can do surgery to remove the cataract. Then at the same surgery they can place an artificial lens or not. If not, one would need to wear glasses or a contact lens to rehabilitate the eye and see.
Observation/surgery. Depending on the type of congenital cataract, the pediatric ophthalmologist may either just observe or recommend eye surgery.