When the Democratic Party had full control of Congress and the Presidency, they were (collectively) very much against illegal immigration[0]. This is why they passed no "reform" legislation whatsoever.

[0] now-Senate Democratic leader Charles Schumer: https://www.youtube.com/watch?v=728SW65tlU0

To give an example, in a 2016 interview Sanders referred to open borders as a "right-wing" "Koch brothers proposal". Trump is worse, is that what you want to hear?

I'm not trying to draw a moral equivalence. But I think the barriers to immigration raised over the last century and more everywhere in the first world are something that can't be blamed on any single party or political movement.