First thing that you will notice is the symptom of discouragement as this will help you to suggest proper remedy. You can identify a discouraged employee if he/she is complaining that there is nothing interesting in the given task. You can also recognise the disturbed person if he/she is sluggish and gets irritated without any major reason.
You should discover the basic reason of this discouragement. Sometimes, they do not want to share so you should ask them in private. Listen to them with patience and try to give the impression that you are not questioning them rather it is just a friendly conversation. You can also take help from HR department if the person is not comfortable sharing with you.
Motivate the discouraged employee by making him/her realise that this situation can affect the progress of the company. Try to stay close to your employees and build personal relationship to keep them motivated.
If you come to know that it is the work load that has brought your employee down then reduce his/her responsibilities. Assign such tasks that are manageable and easy to do. This will give them confidence and inspiration to get back on track.
If the task is beyond their skill level then transfer them to the projects where they feel comfortable. Give them a win-win situation and it will really help them to come out of a troubled period.
Let them know that you are happy with the overall performance. Encourage and appreciate them even for small achievements. This will give them a boost and develop a sense of satisfaction with the job.
If you feel that things are not getting better then you should refer that person to Employee Assistance Plan (EAP).