The literal meaning of the phrase half seas over is halfway across the sea.

This adverbial phrase is first recorded in a letter dated 15^th September 1551, which, on behalf of the King, rebuked the authorities of the ports of Chester and Liverpool for their disregard of the King’s proclamations prohibiting the export of certain specified goods:
—as published in City of Liverpool. Selections from the municipal archives and records, from the 13^th to the 17^th century inclusive (Liverpool: Gilbert G. Walmsley, 1883), by the British antiquary and architect James Allanson Picton (1805-1889):

To my frends the customers, comptrollers and searchers of the King’s Ports of Chester and Lyv’pole
After hartie comendacons having daylie knowledge and experience that by the greediness and covetuous desyre of div’se p’sons, the comodities of this realme that be p’hibited both by p’clamacons and statut^s ben daylie conveyed awaye, w^ch ben taken and retorned againe, when they be halfe the seas over.

The phrase half seas over then occurs in Sir Walter Rawleighs Judicious and Select Essayes and Observations upon the first Invention of Shipping. Invasive War. The Navy Royal and Sea-Service (London: Printed for A.M., and are to be sold by Robert Boulter […], 1667), by the English courtier, explorer and author Walter Raleigh (1554-1618):

Whosoever were the inventers, we find that every age, had added somewhat to ships, and to all things else. And in my owne time the shape of our English ships, hath been greatly bettered. It is not long since the striking of the Top-mast (a wonderfull great ease to great ships both at Sea and Harbour) hath been devised, together with the Chaine pumpe, which takes up twice as much water as the ordinary did, we have lately added the Bonnett, and the Drabler. To the courses we have devised studding Sayles, Top gallant Sayles; Sprit stayles, Top stayles, The weighing of Anchors by the Capstone is also new. We have fallen into consideration of the length of Cables, and by it we resist the malice of the greatest winds that can blow, Witnesse our small Milbrooke men of Cornewall, that ride it out at Anchor, half Seas over betweene England and Ireland, all the winter quarter.

The phrase half seas over came to be used figuratively to mean halfway towards a goal or destination, half through with a matter, halfway between one state and another.

This is first recorded in the following dialogue between Cleomenes, King of Sparta, and Pantheus, a noble Spartan, the favourite of Cleomenes, from Cleomenes, the Spartan Heroe (London: Printed for Jacob Tonson […], 1692), a tragedy by the English poet, playwright and critic John Dryden (1631-1700):

Panth. Sir, will you go?
Cleom. I know not: I am half seas o’er to Death!
And since I must die once, I wou’d be loth
To make a double work of what’s half finish’d.

The phrase half seas over came to be also used humorously to mean half drunk.

This is first recorded in A New Dictionary of the Terms Ancient and Modern of the Canting Crew. In its several Tribes, of Gypsies, Beggers, Thieves, Cheats, &c. With an Addition of some Proverbs, Phrases, Figurative Speeches, &c. Useful for all sorts of People, (especially Foreigners) to secure their Money and preserve their Lives; besides very Diverting and Entertaining, being wholly New (London: Printed for W. Hawes […], P. Gilbourne […], and W. Davies […] – [1699]), by “B. E. Gent.”:

Half Seas over, almost Drunk.
[…]
Suckey, c. drunkish, maudlin, half Seas o’er.

The phrase half seas over means half drunk in a letter published in The Spectator (London, England) of Friday 5^th November 1714—as reprinted in London, for J. and R. Tonson, in 1739:

I have just left the Right Worshipful and his Myrmidons about a Sneaker of Five Gallons. The whole Magistracy was pretty well disguised before I gave ’em the Slip. Our Friend the Alderman was half Seas over before the Bonefire was out.

A folk-etymological explanation of the phrase half seas over occurs in the column Home Help, published in the Evening Post (Reading, Berkshire, England) of Monday 5^th January 1981:

Is there any difference between “three sheets to the wind” and “half seas over” when these phrases are applied to anyone who has been drinking?
“Three sheets to the wind” originally meant that most of the ropes (or sheets) used to control a sail were flapping in the wind and that the sail was almost out of control; hence it was applied to those who were drunk and unsteady or staggering.
“Half seas over” is thought to be a corruption of “op-zee-zober” or “over-sea beer”, a very strong English beer imported into Holland and the phrase is used to indicate a person almost dead drunk or helpless.