A friend of mine booked a trip on a cruise ship for a crossing.
I forget what it’s called, but when cruise companies need to move a ship from one port to another (like it’s getting too cold for Alaskan cruises, so the ship will be moved to the Caribbean) you can book a cabin on it. It’s the same experience as a normal cruise but with no port stops.
Or a new ship has been built in Italy and needs to come over to the US. You fly to Italy and stay on the ship while it crosses.
I have friends who recently took the Queen Mary 2 from the UK to NY, because they were relocating and didn’t want to put their dog through the trauma of a flight.
Yes, you can take dogs. Yes, they stay in your cabin.
There’s a kennel on board too. Cute video -
The term of art for this is a repositioning cruise.
It’s hard to answer the OP definitively, but I think this is the best answer – 1974.
The Comet airliner in 1958 (mentioned above) is a good answer for “the beginning of the end.” So you basically had a sixteen-year period when it wouldn’t be odd or unusual to go either way.
I took the QE2 from New York to France in 1980, and based on talking to people on board who had been doing it for years, it was not that much different from the past. Certainly more formal than cruises today, but that was more the time. And no port tours in the middle of the Atlantic. But a day on the ship was not that much different from days at sea which I’ve had during Mediterranean and Baltic cruises.
Hint: If you do a transatlantic cruise, do it East to West. You gain an hour every night, and you need it to eat. 
I think the QE2 did do regular Southampton - NY runs up until she was replaced by QM2 (here’s her 1996 itinerary, for example) There might have been some years where that route wasn’t used eg 1982 when she was used as a troop ship for the Falklands conflict, and a few years later when she needed an engine replacement. I can’t find a complete list of her sailings though.
Although one thing to keep in mind is that, somewhat ironically, the arms race of ever faster, ever more luxurious ocean liners from the 30’s through the 50’s was largely a symptom of a business in a long decline. What killed it wasn’t an alternate mode of transport, but the immigration restrictions that were passed in the US in the mid-20’s that effectively ended the 2nd and especially 3rd class passenger trade had been the bread and butter of the great ocean lines. That along with bad economic conditions in the 30’s led the few lines that didn’t merge or go bankrupt to have to have to compete for an ever-shrinking pool of well-to-do passengers.
It’s true that the airlines were certainly the final nail in the transatlantic passenger lines’s coffin, but I think the story of the Jet Age is less about the airlines stealing what wealthy customers the ocean liners still had but rather about the creation of a whole new market of middle-class international tourists and business travelers. By the time international flights started getting more affordable in the 70’s, the average jet passenger wasn’t someone who would have gone by ship a generation earlier; they were someone who wouldn’t have been able to travel internationally at all.
It also was “the end” for a number of people, at least until they fixed the stress fracture problem.
I wouldn’t be surprised if some people still do the North Atlantic crossing for relaxation and nostalgia, as well as luxury accomodations. And maybe food (the France had legendary cuisine).
Regular in the year-on-year sense, but not frequent within the year - transatlantic runs were a minority activity, interspersed by often months and months of other itinerary such as world or Med cruises.
The Queen Mary is most certainly NOT a cruise ship. It is an ocean liner. Saying it is a cruise ship is like calling Motel 6 the Waldorf. Cruise ships can’t sail around the world. They just aren’t designed for it.
Interestingly, the Titanic II makes it’s first voyage in 2018. It’s modeled after the doomed ocean liner albeit a little larger and much safer.
Missed the edit window. A cite.
The QE2 (1964-2008) was probably the last “Real” “Oceanliner”.
Cunard (which acquired White Star (RMS Titanic) in 1934) is now owned by Carnival.
QM2 is now an over-sized cruise ship (see any current pic).
So, for OP: 2008, last voyage of the QE2.
The “classic” oceanliner did not have 10 decks. The QM2 may have begun life as a “First, Second. and Steerage” ship, but its current form is patterned after cruise ships.
A ++1 for username / post content? (Or is this one just a little too obvious?) 
Typo in title fixed.
Continued longer between Aus and England. I think the last migrant ship to arive in Australia was the SS Australis, December 1977.
Could you clarify what you mean by that? Is a cruise ship vs. ocean liner somehow defined by appearance?
Fear of flying is why my mom took herself, my older sister and me to England via the sea in 1972, while Pop went via air. (He couldn’t take the time off from work.) We went via the QE2 to England, something I vividly remember as being awesome despite being only 5 because the crew and passengers were so amazingly lovely.
Alas, we returned via the France, which was a far less enjoyable experience for me; snotty crew and terrible food in the children’s area that I couldn’t keep down. (And it wasn’t seasickness, since the seas were way choppier on the QE2 and I loved it.)
The France was rechristened the Norway, if I recall correctly, which eventually had some kind of accident… fire? But continued for a few years into the 2000s.
I will definitely be taking the Queen Mary 2 some day. That transatlantic journey of my youth was one of the most delightful experiences, all these years later, and though I doubt I can repeat the magic, being old and wizened and all, I’m sure gonna try.
Anyway, point is, having religiously read all the literature that Cunard sends me, I can assert (as does Cunard) that the QM2 is most definitely an ocean liner, and in fact is the largest passenger liner ever. Her design may be lavish but for the cost and the length of the journey, why on earth would you expect anything but grandiosity?
Cunard’s European cruise ships are the Victoria and the new Elizabeth 2.
(In all honesty I’m rather surprised it took them so long to come up with a Victoria. Why would they go with Mary, of all awful sovereigns?! At least with Mary 2 I can convince myself it’s named after William’s wife, not Bloody Mary.)
Cunard’s first Queen Mary was named after the consort of George V. Queen Mary 2 is not named after any queen, but after the earlier ship.