Can a baby get a cold sore?

Yes: Babies also can get sores.

Cold sores: Babies are also suspectible to the Herpes simplex virus and can get cold sores.

Fever blister: Fever blister is pretty much universal. We get it mostly from our parents. Is caused by herpes virus 1. Is alway in your body, goes in sleeping mode and gets activated at random (some people never, other people more often). Triggers like a cold, stress may activate it. Talk to pediatrician because we want to be sure is a cold sore and not something else. We want to treat patient the proper way.

Cold sores: Cold sores can be spread by touching cold sores (for instance if person with cold sore touches it with their hand - the herpes simplex virus can be spread further from there), kissing, applying chapstick or coming in contact with other personal items that have been contaminated.