Trans-Atlantic cruises are not common. Cunard has regularly-scheduled trans-Atlantic crossings between Southampton and New York City. It is not considered a budget cruise line, however, Other cruise lines offer trans-Atlantic cruises during Spring and Fall when they move ships between the Mediterranean/Europe and the Caribbean.
As for a website that offers comparisons and detailed information, you may have an easier time of this by contacting a travel agent.
Personally, I traveled on a trans-Atlantic cruise between South Africa and Chile, and found it to be enjoyable. However, you'll need to be the kind of person who doesn't mind four or five or seven or more days at sea without any ports-of-call.
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