Once your Aide has reached 50 Trust, you may equip them with a ship. However, like yourself, your aide can only sail a ship that is allowed for their levels.

To equip your aide with a ship, you must first make them your Aide Captain by going to Character -> Change Aide duties.

In the upper right-hand corner, you must click the button that says "Assign Aide Captain." Then to equip the ship on your Aide, go to Ship/Fleet -> Aide Ship Info.