There is a relatively simple and cheap option for those who don't want the rigours of a freighter and don't have the skills to crew a yacht.
That is to take a repositioning cruise on a normal cruise liner.
These get sold off very cheap, since it's basically fifteen days on a cruise ship without stopping at the interesting islands and tourist spots that happen on a normal cruise.
A fifteen day cruise (one way) can set you back less than the price of a (return) transatlantic air fare - and you get meals and entertainment with that.
However they tend to be at only certain times of the year, and the American end will be in the Caribbean, not New York (mostly).
Here is the trans-atlantic repositioning schedule for Royal Carribbean .
Here is a more general article .
Now you know they exist you should be able to find them for any cruise line you want.
Any cruise line that operates in both the Carribbean and Mediterranean will do a few of them a year.
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The two main options that I am aware of are: The Queen Mary 2 , the only regular transatlantic passenger line, Freighter travel , which is not cheap either.
I would not be surprised if you could join a crew aboard a luxury yacht in Monaco going to the Caribean for the winter.
For example, Cruisers forum has a Crew wanted section.
In that case, you get paid for the experience.
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Mouvicels answer is correct.
I'd like to add a bit more on freighter travel.
Although I have no personal experience with it, I think it's an interesting option that I would certainly like to try out some day.
The information below is fully based on what I have read on the internet.
In freighter travel, you get to travel on-board a freighter.
Those freighters take only a handful of passengers, typically a dozen or so.
You need to entertain yourself.
The freighter has a particular route and you can join either the whole way, or from port to port.
You may need to pay a not inconsiderable amount of money to use such an unusual port of entry in the United States.
Your luck is that transatlantic freighter travel is one of the most popular routes.
You might pay around US$1500.
This cannot be directly compared against an airfare, because for this price includes all meals and lodging.
For some routes, the prices are in US$/day plus a base fee.
It will probably take between 8 and 12 days.
Some sources for freighter travel: Wikivoyage on freighter travel .
It countains a lot of practical information and further links.
An overview of trans-atlantic routes .
You have the choice between the North Atlantic (e.g. UK - New York), the Mediterranean (southern Europe - USA), or to travel to the Caribbean.
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My website loco2.com has an informative post on cargo-ship (freighter) travel .
The post recommends various websites for how to book, including Strand Travel in the UK, which has an extensive range of voyages listed on its website , including Southampton (UK) to New York for around £1,000.
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Although it perfectly answers your question, I am sure this won't make your cut...
But if you truly are a backpacker, then you can actually traverse the seas on human power.
This idea comes from the recent news of Jason Lewis, who circumnavigated the world only on human power !
Source: http://www.bbc.co.uk/news/magazine-21377623
Now that's just baw's of steel, so I explicitly forbid you to undertake it and thus have no liability whatsoever!
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A Cunard ship is comparable in price to any of these and cheaper if you take an inside stateroom.
And the food is better although the view will be the same.
I've done it
and it's fabulous.
Would do it again in a heartbeat.
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Suppose that:   I have both valid US & Schengen visas  I am a savvy backpacker with a preference for economical travel  I have no fixed schedule to meet. A longer travel time is even better.  Any European port (of Schengen country) is okay.
How can I cross the Atlantic by ship from Europe to New York? (I am not talking about luxury passenger cruise-ships.)
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<usa><europe><schengen><sea-travel><ships>
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You can take a Cunard cruise on the Queen Mary 2, the only regular transatlantic passenger line, or join a freighter on part or all of its route. Finally, you may be able to join a crew of a luxury yacht going to the Caribbean for the winter. None of these are cheap, but Cunard will be cheaper if you take an inside stateroom. For those wanting to cross the Atlantic by ship, freighter travel is recommended. The advantage is you can join the ship at any point on its route. Alternatives include a Cunard ship or joining the crew of a luxury yacht, heading to the Caribean for the winter.
There are two ways to cross the Atlantic by ship: the Queen Mary and freighter travel. You can join a freighter at any point of its trip. Cross the Atlantic as crew onboard a luxury yacht travelling to the Caribean for the winter. Cunard ships are an excellent way to cross the Atlantic.