Can my 1 year die after possible exposure to cold sore after touching band aid used to cover it possibly the cotton part nd putting finger in mouth?

Extremely unlikely: would be more likely to die in a car wreck

Babies from birth to: 6 weeks are at risk for Herpetic Encephalitis if they become infected with Herpes Simplex 1, although caution dictates advising one's pediatrician if a baby under 3 months gets a cold sore. Primary infection in a 1-year-old can cause gingivo-stomatitis , inflammation & ulcerations of mucous membranes of the mouth, gums, tongue & hard & soft palate that lasts about 2 weeks.