When were passenger ships from England/Ireland to America the most popular?

They were very popular before the age of passenger aircraft became the normal way to travel. Flying anyplace in the early days were only for the very rich, it was not untill the late 50,s that flying started to become the way to travel also for the very rich. Before that it was by sea, by sea you had massive passenger ships, and with different prices to sail in them, first class down to steerage class which was the cheapest fare . If you watch the movie Titanic that would give you a good idea of the class of travel in those days, on passenger ships. Fascinating times they were.

After the Civil War up until World War one. It the spiked again between the two (but not during the depression). There was a lot of movement before the Civil War, but most of it dropped off because of the hostile climate at the time. As World War 1 hit, the United States was a haven of prosperity, and people filled in insanely. During World War I, a bit of scarcity hit as its hard to travel during a heavy war (though a lot of refugees definitely came over during the first and second World War). After World War I, many people came across once more from Europe, but much of that was Italian and Slavic immigrants.