It sounds like your dog might have either kennel cough
or canine influenza
Both of these are very easily picked up from another dog.
While coughing is common with these upper respiratory diseases, much like a cold in humans where some people cough and others just sneeze dogs can vary in the symptoms depending on the type of virus they caught.
You can see if some Robitussin DM will help the dog breathe better and control any cough or sneezing or runny nose. Read here about cautions and dose info
You can see and hear a dog with kennel cough here
If you see a colored discharge from nose or eyes, if your dog coughs stuff up, seems lethargic, has any real trouble with breathing, or goes off eating, see your vet as your pet may have a secondary bacterial infection that will need antibiotics or other meds.
You can read more about nasal sneezing and discharge here
Hope this helps you!
The information is exact if you read the links.
I can tell you how to judge if the dog needs a vet but you have to make the decision. A viral infection cannot be treated by antibiotics but a secondary bacterial infection can be.
If you read the information on Robitussin DM use you will see its dosed in mg by weight of the dog so you give the correct amount.
If the Childrens product has NO other ingredients in it than the adult formula has then it might be OK but usually a correct dose of the adult formula is a smaller volume to get the dog to swallow. For example you might need 1/4 teaspoon of adult formula or 1 1/2 teaspoons of the children's based on mg required per pound. Its much easier to get the 1/4 teaspoon amount into the dog.