It is used to calm down the Hardman Norns when they feel particularly aggressive and begin to take their anger out on their surroundings. Leaving a few around the ship can train your Hardman Norns to drink it when angry.
In addition to decreasing anger, it increases sleepiness and tiredness, so it can be used as a sleep aid.
MNB has created a Calm Balm Vendor which you can install in your world for ease of dispensing Calm Balm. Steelfoot created a Giant Calm Balm which also dispenses calm balms. Alex created a Calm Balm Vendor which is a smart vendor.