My initial impression was just that there were always more people in favor of a decrease than an increase, but when I graphed the data I realized that there was a definite pattern: more support for a decrease until the mid-1990s, and then increasing support for immigration after that time. There are some ups and downs, notably a drop in support for increased immigration after 9/11, but the trend after the mid-1990s is clear.
This figure shows the percent favoring an increase:
Before the 1990s, almost all of the change in the average involved a shift from staying the same to decreased--after that time, much of it involved a rise in the number of people saying it should be increased.
As a Thanksgiving bonus, here are some pre-1965 questions on immigration:
Roper/Fortune Survey [November, 1947]
Would you vote yes or no on a bill in Congress to let 100,000 selected European refugees come to this country in each of the next four years, in addition to the 150,000 immigrants now permitted to enter every year under our present quotas?
18% Yes
72% No
5% Depends
5% Don't know
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Foreign Affairs Survey [April, 1955]
In general, do you think the United States is letting too many immigrants come into this country, or not enough?
39% Too many
37% About right (vol.)
13% Not enough
11% Don't know
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Hopes And Fears [September, 1964]
Do you think the number of immigrants allowed to enter the U.S. each year should be increased somewhat, decreased somewhat, or kept at about the present level?
6% Increased
38% Decreased
46% Kept at present level
10% Don't Know
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Harris Survey [May, 1965]
President (Lyndon) Johnson has proposed that the immigration laws of this country be changed to allow more people into the United States as immigrants. From what you know or have heard, do you favor or oppose letting more people come to the United States as immigrants?
24% Favor
58% Oppose
18% Not sure