To help soothe a cough in a child under 1 year old, it is recommended to focus on nasal secretions that may be causing the cough. Using a simple syringe without a needle and following the technique described in the provided document can help with this. Additionally, placing a wet and warm cloth with Vicks™ near the child's bed can provide some relief. If the cough persists for more than 2 weeks, it is advisable to consult a doctor.