Caroline Alexander explores the ancient question:
This wine-dark sea has haunted many imaginations. Nineteenth-century British prime minister William Gladstone posited in Studies on Homer and the Homeric Age that the Greeks had a form of color-blindness, their optical palette limited to black and white, and possibly red. Another theory was that a type of algal bloom, red tide, made the Homeric-era Aegean wine-red. In the 1980s, the view was advanced that since the ancient Greeks mixed their wine with water, the alkaline water common to the Peloponnesus would have turned red wine to blue. Another view came from a retired classicist who watched “an unusually vivid sunset” over the sea at the mouth of the Damariscotta River, in Maine, on an evening when the sky was filled with ash that had floated east from the eruption of Mount St. Helens, and was struck by the color of the sea “reflected in the outgoing tide of the dark estuary.” The sea, he said, was the color of Mavrodaphne, a wine of deep purple-brown hue, and the epithet for Homer’s wine colored sea, he speculated, meant “sunset-red.”
Finally, many contend that the phrase is meaningless, an empty expression with a poetic ring whose purpose is only to fill out the metrical requirements of a line of the verse.
Update from a reader:
Of course, one further explanation is that Homer was merely making an allusion to the opaqueness of the sea, not its color per se. He was comparing the inability of light to pass through wine (because of its color) with the similar effect caused by the depth of the sea.
Radio Lab did a piece on this a while back. One of the possible explanations is that at that time there was no equivalent word in Greek for blue.
(Painting: Ulysses Deriding Polyphemus by J.M.W. Turner)