The Black Sea during Antiquity had an ambivalent reputation, being considered as both "inhospitable" (Ἄξεινος) and "hospitable"(Εὔξεινος). This is not particularily surprising as sea, in general, was considered a sphere carrying both wealth and dominance, but as well wild, changing and making one's pay for its benefit. Depending of one's relation and history with the Black Sea, a different picture could be made.

Aeschylus makes Prometheus caling this sea (near Salmydessos) a cruel mother (PB 725-727), after noting that you're not in Kansas Europe anymore. The Black Sea was well known of Athenians and dear to their trade interests, so it's not just a manner of speech for "this is far and unknown" but it highlights its "foreignness" giving the land a personality.
In other plays, Aeschylus mention the lot of shiprwrecks, a lack of safe harbours protecting ships from a harsh northern wind, and Barbarians less than well-meaning when it comes to welcome sailors rejected by the sea. He wasn't the only one using these tropes, and Sophocles uses them too.

The issue is that other authors in the same time made a quite different picture : a sea with wealthy coasts, with a lot of coastal rivers one could take shelter. There's one big exception, underlined by the choirs in Antigon : the northern wind, violent, boiling sea in its very depths. That was what could make Black Sea a trap for marineers.

Of course there's as well a bit of tragic tropes in these theatrical depictions, and what classical Athenians might think of the Black Sea was probably more mixed : a good part of their trade, especially in grain and prestigious Barbarian goods came from the region, at the crossroad of the Crimean/Volga and Black Sea trade ays (the same reasons why Genoans were interested in the regions still after a thousand years) making the regions of the Black Sea both familial enough, and still Barbarian and host of several litterary tales (Amazons, Colchis, etc.) known in mythological narratives.