If your 1-year-old has been having a persistent cough and cold since infancy, it is important to get a pediatric evaluation, as it could be due to allergies, infections, or other underlying conditions. Home care includes keeping him hydrated, using a humidifier, and gently clearing nasal mucus. Avoid over-the-counter cough medicines for children under 2 unless prescribed. A paediatrician can recommend the right treatment or medicine after checking the child's health and identifying the cause.