The old port in Fira is accessible only:
By cable car: It is perfectly safe, it consists of 6 wagons and has capacity of 6 persons per wagon. During high season (April-October) it runs every 20 minutes. Single tickets cost €5 for adults and €2.50 for children.
On foot: You would need to walk down the famous 600 steps of the traditional road that connects the old port with the town of Fira. The descend takes about 20-30 minutes, depending on your physical condition.
By donkey: The donkeys used to be the traditional transportation way of the island. They go up and down to the old port of Fira frequently. A single ride costs around €5 per person and takes about 20 minutes. Most people prefer to take the donkey on their way up.