"The Argonauts sailed on in gloom. The Seirenes  were behind them, but worse perils lay ahead, at a place where two seas met and shipping came to grief. On one side the sheer cliff of Skylla  hove in sight; on the other Kharybdis  seethed and roared incessantly; while beyond, great seas were booming on the Wandering Rocks."
-- Apollonius Rhodius, Argonautica 4. 921

A map of the wanderings of Aeneas, showing the positions of Charybdis and Scylla (underlined in red), from an edition of The Aeneid of Virgil, Book III, edited by Philip Sandford, London: Blackie & Son. 1900

Some of the greatest heroes encountered the terrifying and destructive force of Skylla and Kharybdis on their travels: Odysseus had to pass them twice, and the Argonauts came upon them as well. We will discuss Skylla at a later date, but for now, we will discuss Kharybdis--monster and Goddess--who controls the tide.

The ancient hellenes in the time the Odysseia was written seem to have been largely unaware that the rise and fall of sea levels are caused by the combined effects of the gravitational forces exerted by the Moon and the Sun and the rotation of the Earth. Instead, they held a single entity responsible: a monster or a Goddess controlling a whirlpool, as described by Hómēros:

"Odysseus, you will notice the other cliff is lower, only a bow-shot away, and a great fig-tree with dense leaves grows there. Under it divine Charybdis swallows the black waters. Three times a day, she spews them out, and three times darkly sucks them back again. No one, not even Poseidon, could save you from destruction if you are there when she swallows." [Bk XII:36-110]

For those of you who grew up with huge differences between high and low tides, it might be hard to grasp how the ancient Hellenes attributed such a difference to a single whirlpool entity. Well, they didn't; Even modern Hellas hardly has a difference between high and low tide; a foot or two at max. As such, it becomes a lot more feasable for that amount of water to be swallowed and regurgitated.

The development of tidal science began in antiquity with the cosmology of Aristotle (384 BC – 322 BC), a Hellenic philosopher and polymath, a student of Plato and teacher of Alexander the Great. In researching this subject, I keep happening upon a quote by him that I can't place in any of his works, but would love to have someone source for me. It goes:

"It is even said that many ebbings and risings of the sea always come round with the Moon and upon certain fixed times."

The closest I have gotten is from the 'Meteorology':

"Again, most earthquakes and the severest occur at night or, if by day, about noon, that being generally the calmest part of the day. For when the sun exerts its full power (as it does about noon) it shuts the evaporation into the earth. Night, too, is calmer than day. The absence of the sun makes the evaporation return into the earth like a sort of ebb tide, corresponding to the outward flow; especially towards dawn, for the winds, as a rule, begin to blow then, and if their source changes about like the Euripus and flows inwards the quantity of wind in the earth is greater and a more violent earthquake
results." [Bk 2, Pt 8]

"Every one admits this, that if the whole world originated the sea did too; for they make them come into being at the same time. It follows that if the universe is eternal the same must be true of the sea. Any one who thinks like Democritus that the sea is diminishing and will disappear in the end reminds us of Aesop's tales. His story was that Charybdis had twice sucked in the sea: the first time she made the mountains visible; the second time the islands; and when she sucks it in for the last time she will dry it up entirely. Such a tale is appropriate enough to Aesop in a rage with the ferryman, but not to serious inquirers. Whatever made the sea remain at first, whether it was its weight, as some even of those who hold these views say (for it is easy to see the cause here), or some other reason-clearly the same thing must make it persist for ever. They must either deny that the water raised by the sun will return at all, or, if it does, they must admit that the sea persists for ever or as long as this process goes on, and again, that for the same period of time that sweet water must have been carried up beforehand. So the sea will never dry up: for before that can happen the water that has gone up beforehand will return to it: for if you say that this happens once you must admit its recurrence. If you stop the sun's course there is no drying agency. If you let it go on it will draw up the sweet water as we have said whenever it approaches, and let it descend again when it recedes. This notion about the sea is derived from the fact that many places are found to be drier now than they once were. Why this is so we have explained. The phenomenon is due to temporary excess of rain and not to any process of becoming in which the universe or its parts are involved. Some day the opposite will take place and after that the earth will grow dry once again. We must recognize that this process always goes on thus in a cycle, for that is more satisfactory than to suppose a change in the whole world in order to explain these facts. But we have dwelt longer on this point than it deserves." [Bk 2, Pt 3]

In general, I would say the ancient Hellenes were aware of the connection between the Moon (and Sun) and the tides, but neglected to make a causal connection. For a long time Kharybdis was the most logical of options. Another name of hers is 'Trienos' (Τριενος), 'Three Times a Day', alluding to the earlier stated behavior of the whirlpool. Suidas, by the tenth century AD, had placed the actual whirlpool with accuracy. This location is portrayed on the map above:

"Kharybdis (Charybdis): It sucks up the sea around Gadeira and furiously spirals around again. It is said that it all leads down to chaos and destruction. Priskos (Priscus) says about Kharybdis: ‘They sail by Sikelia (Sicily) in front of Messene and by the strait of Italy where Kharybdis [is], taking in stormy winds, and sucking those men in. Kharybdis and Skylla (Scylla), lying in a narrow place, are subject to the currents of the oceans and sink those sailing past. There Odysseus lost all his companions with the ships; he himself was carried away hanging on to a board in the currents of the sea.’"
It is easy to link our modern understanding of tidal physics to the minds of the ancient Hellenes, but I am in no way certain they actually linked the tides to the Moon and Sun. As such, neither Selene, nor Artemis, nor even the Oceanic Gods like Poseidon would have had any influence on the tides: they were in the domain of either another divinity, or a monster, and both were out of reach.