Both parties are anti-immigration in practice. The only difference is in their posturing to their respective bases around election time.
Consider that the last major amnesty was under Reagan and the last major tightening of immigration rules was the IIRAIRA under Clinton. The modern deportation machine was really spun up by Obama - he removed more people from the US than any other president, almost 1% of the entire US population was deported by Obama. 50% more than Dubya and more than Trump. [1]
It's been two years under full-on Democratic party rule, the remain-in-Mexico policy is being walked back but still in effect. [2] Children are still being separated from their parents at the border. [3] [edit] The public charge rule still exists, but was returned to the classical definition. [4] Indian-born folks are still in 50+ year queues to get green cards subject to deportation at the whims of their employers. Consulates abroad still have year-long backlogs for appointments to get visa foils so people here, legally, in the US, cannot leave the US as they wouldn't be able to get back in without a new foil. I have friends who haven't left the US in years to see their families.
More of the US-Mexico border wall was built under Obama/Biden than it was under Trump, and Obama was behind the implementation of the biometric exit control program.
[1] https://www.cato.org/blog/deportation-rates-historical-persp...
[2] https://www.cbsnews.com/news/judge-suspends-biden-administra...
[3] https://www.vera.org/news/children-are-still-being-separated...
[4] https://www.dhs.gov/news/2022/09/08/dhs-publishes-fair-and-h...
https://en.wikipedia.org/wiki/Congressional_Progressive_Cauc...