That looks like a positive trend--the correlation between the average and time is .57, with a p-value of .03.
A second question was asked twice in 2006, once in November 2015, and once in June 2016: "do you favor or oppose...deporting all illegal immigrants back to their home countries?" The percent in favor was 50, 47, 52, and 32. It's hard to interpret that, since the figures for the last two surveys are so different, even though they are just six months apart. Still, it suggests a move away from support for deportation.
Finally, one asked if you "favor deporting as many as possible or do you favor setting up a system for them to become legal residents?" That was asked in 2007, 2009, 2010, and several times since 2015.
That looks like a definite decline in support for deportation. However, in 2007-2010 the question started with "If it were possible to locate most illegal immigrants currently in the United States, would you"; in 2015-6 it started with "What do you think should happen to the illegal immigrants who are currently working in the United States." That change might account for some or all of the difference between the early and later years--people may be more favorable about immigrants who are "currently working" than immigrants in general. But there does seem to be a sharp drop in support for deportation between 2015 and 2016, as with the second question.
None of these questions gives definitive evidence, but taken together they suggest that opinions on policy towards illegal immigrants have moved in a liberal direction in recent years, which raises an obvious question: how do you square that with the success of Donald Trump? I will take up that question in a later post.
[Data from the Rope Center for Public Opinion Research]