How dangerous is second hand smoke for the baby?

Very: Second hand smoke increases the risk of sudden infant death, asthma and many other diseases.

Doubles illness: Second hand smoke doubles the frequency of an infants respiratory illnesses. This includes middle ear infections, bronchitis and susceptibility to complications of RSV .This includes the well meaning but ineffective effort to "just go outside" to smoke. If there's a smoker in the house/day care/church nursery/elevator your baby will carry the particles in its nose for weeks at a time.

Quite!: Very dangerous, & my colleagues posted good answers. Smoking is dangerous to babies before birth: prematurity and low-birth weight. Afterwards, the smoke soot is electrically charged and attracts allergens which can then get into the lung and more easily cause allergies. Babies get chronic ear infections, far more frequently, and recurrent infections can be associated with hearing loss!