Reading supplementary material related to the Iliad (notably The Trojan War: A New History, by Barry Strauss; I'd also have to check but I'm certain the introduction by Caroline Alexander in her translation of the Iliad also mentions this) I noticed that a very common theme was that historical Troy was always talked about as controlling travel of ships between the Aegean and Black Seas (or at least taxing anyone who docked there before undertaking the journey) which is what made the city so prosperous.
Recently I watched a video Great Voyages: Jason and the Golden Fleece by Brian Rose where he says that they haven't found any evidence of travel/trade in the LBA between the Aegean and Black Seas; the link to the video above is the timestamp to when he talks about this. Earlier in the video he says that the Bosphorus wasn't navigable by bronze age ships and goes on to say that it hadn't been navigable until 8th century BC.
So my question boils down to whether this is true or not. Is there really no evidence of any maritime travel or trade between the two regions? Everything I've read about the late bronze age seems to indicate that trade between societies was quite prolific so it seems odd that there wouldn't be travel between the two seas. Especially when every map of the Hittite empire I've seen shows it straddling both seas and travel by boat was (at least everything I've read says this) significantly easier and faster than over land.
First of all, regarding that video, trying to pin a Late Bronze Age date on a Greek myth is a bad idea. (And doing so based on the Argonautica doubly so.) The ancient Greeks believed that there had been an “Age of Heroes”, but they had no idea of what the "Bronze Age" was. Anyone who wants to argue that the Greek myths reflect the conditions of the Bronze Age usually ignores the fact that there’s several centuries that separate the Bronze Age from the time when the first stories get written up (or even represented in art), and it ignores the fact that the ancient Greeks did not have professional archaeologists who could reconstruct what the past was like. See my answer here for details.
Was the Bosphorus not navigable in the Late Bronze Age? Depends on your definition, but ships could, as far as I know, pass through. The main point is that going through the Dardanelles wasn’t trivial. Thomas F. Tartaron, in his book Maritime Networks in the Mycenaean World (2013), provides a historicizing interpretation -- how fitting! -- of the “Clashing Rocks” from the Argonautica.
He writes (p. 136):
These two floating masses of rock crashed together whenever a ship tried to pass between them, demolishing the ship and its crew. The origin of this story may lie in the difficulty of passing from the Dardanelles through the Bosporus into the Black Sea, because of the combination of prevailing northerly winds and the strong current of the Black Sea outflow. Ships could only make headway by waiting for a strong southerly/westerly wind to propel them against the current. The danger of being driven against the rocks if the wind failed may have prompted the story of the clashing rocks, but it is easy to see how such an apparition could be inspired by any turbulent narrows.
There isn’t much evidence for Mycenaean trade with the Black Sea region. In The Cambridge Companion to the Aegean Bronze Age (2008), edited by Cynthia Shelmerdine, Christopher Mee surveys the evidence in his chapter "Mycenaean Greece, the Aegean and Beyond" (pp. 362-386; esp. pp. 369ff). He mentions Troy, where large amounts of Mycenaean pottery have been found. "Surely," he writes, "the location of Troy on the Dardanelles prompted these contacts. Contrary winds and currents would often have made the straits impassable, especially in the summer. Ships that took shelter in Besik Bay must have been a lucrative source of revenue for the Trojans" (p. 371). But other possibilities may also have factored into this, such as trade directly with Troy or with the Anatolian hinterland; there is little proof -- and a lot of assumptions! -- that Troy actively exploited the sea trade through the Dardanelles.
Askold Ivantchik has gone even further and states outright that there were no (direct) contacts between the Mycenaean world and the Black Sea. He wrote a chapter on the earliest contacts between the Greeks and the Black Sea in The Northern Black Sea in Antiquity: Networks, Connectivity, and Cultural Interactions (2017), edited by Valeriya Kozlovskaya.
He notes (p. 8):
The few objects that had been interpreted as evidence for contacts between the Pontus and the Aegean have turned out to be locally produced, and even if they show some similarity to artifacts manufactured in the Aegean , we can talk here only about cultural influences resulting from overland contacts through many intermediaries, and not about direct imports. In the early 1990s, Stefan Hiller tried to revisit the hypothesis of the existence of regular maritime contacts between the Mycenaean civilization and the Black Sea region and collected available archaeological data that allegedly supported it. However, this data actually showed the weakness of his arguments -– none of it can withstand critical examination.
What little evidence there was most likely transported over land rather than by sea, and was traded via e.g. Central Anatolia. Ivantchik may be a bit too skeptical about the overseas contacts, but it’s true that the evidence is certainly not overwhelming. It could also be, of course, that what trade there existed may have left little in the way of evidence that is archaeologically visible (e.g. textiles, foodstuffs).