My toddler was prescribed an inhaler to relieve a cough that doesn't go away but I was told that excessive use of inhalers can develop into asthma?

Not quite. The inhaler is to relieve bronchial constriction. The inhaler will not lead to asthma....Asthma leads to the use of an inhaler. Excessive use of any medication can lead to unpleasant side effects that can land you in the er. Use the inhaler properly and as directed, and if there is still a problem with the cough, be seen again.