So I suspect it was not what you meant by "Age of Sail", but given the term is ambiguous and the Romans had sails, I will say that, yes, passengers getting rides on ships was certainly something that happened, and common enough for a bit of maritime law to be dedicated to it and the situations that might arise from it. I'll mention the legal situation to talk about what this may have looked, but to lead with a gee whiz wow I'll give a particularly cool example from Lucian's Alexander the False Profit (which was often thought to be made up but isn't, but textual criticism of Lucian is beside the point)(important part in bold):
- Indeed, he seriously imperilled one of the Epicureans who ventured to expose him in the presence of a great crowd. The man went up to him and said in a loud voice: “Come now, Alexander! You prevailed upon such-and-such a Paphlagonian to put his servants on trial for their lives before the governor of Galatia on the charge that they had murdered his son, a student at Alexandria. But the young man is living, and has come back alive after the execution of the servants, whom you gave over to the wild beasts.” What had happened was this. The young man cruised up the Nile as far as Clysma, and as a vessel was just putting to sea, was induced to join others in a voyage to India. Then because he was overdue, those ill-starred servants concluded that the young man either had lost his life during his cruise upon the Nile or had been made away with by brigands, who were numerous at the time; and they returned with the report of his disappearance. Then followed the oracle and their condemnation, after which the young man presented himself, telling of his travels.
So not only could you hitch a ride, you could hitch a ride to India.
Unfortunately the details of how these things happened is a bit scanty. Most of the literary accounts of sea travel we have are describing people on official business or who essentially own the ship. When Caesar was famously captured by pirates, he was possibly traveling in a sort of semi-official capacity, and even if he wasn't he was hardly representative of the average traveler. There is also a bit of the Satyricon that takes place on a ship, which is often a great source for daily life, but the details of how they got the boat are perhaps in a missing section. The details of he journey are also not so much on the traveling conditions as the weird interlocking sexual histories of the passengers, but there are two tidbits: one is that the characters slept on the deck, and the other is that when the storm inevitably came, one of the characters made off in the ship's boat. Anyway, leaving that aside.
Law codes give us another glimpse, at least insofar as they make accommodations for a given ship having multiple different owners for cargo. There isn't all that much to say about this, but it gives a sense that a given merchant ship would not necessarily be in the service of a single owner. In fact, the way that the title of naukleros (ship owner) shows up in inscriptions implies that ship owners did not necessarily own cargo at all, one particular striking example is a dedicatory inscription by Aelia Isidora and Aelia Olympias, two sisters (you read that right) who identified themselves as both merchants and ship owners. Taking these little bits of information I think we can say that there would not necessarily be particular types of ships that are for carrying passengers, but rather that cargo ships could easily take on passengers as another sort of cargo, as the situation of maritime law allowed for these sorts of divisions within a ship's hold.
So who would be doing this sort of travel? ORBIS gives a trip from Alexandria to Rome as about 500 denarii, using the prices of Diocletian, so that is basically equal to five modii (pecks) of wheat. This is a decent amount but it isn't crazy or anything, so while obviously the poorest would not be able to make the travel the middling sort would. And, delightfully, we actually have something of an example: in the late first century CE, a woman named Hikane wrote a letter to her son, who was working as a "harbor man" (unclear what this mean, but hardly an elite position!), begging him to come home or at least write more. She was from "Arabia" (probably the Levant) but the letter was written from Berenike, a port along Egypt's Red Sea, which is a wonderful example of non-elite (and female!) travel. It is also a great example of travel for personal reasons--not pilgrimage, not business, not official duties, just a woman trying to see her son.
So unfortunately I can't go into much of what the travel experience was like, we can say with some certainty that it happened for all sorts of reasons, to all sorts of places, and with all sorts of people.