Democrats have done exactly that since 1965 with the Hart -Celler Act. Prior to that there was a quota system that had as it's goal not to alter the existing demographics of the United States.
The
National Origins Formula was an American system of
immigration quotas, used between 1921 and 1965, which restricted immigration on the basis of existing proportions of the population. It aimed to reduce the overall number of unskilled immigrants (especially from
Southern Europe and
Asia), to allow families to re-unite, and to prevent immigration from changing the ethnic distribution of the largely
Protestant Northwestern European-descended United States population.
National Origins Formula - Wikipedia
The Hart-Celler Act changed all of that, and began favoring immigrants from Latin America, Africa, and Asia. We know what effect that has had on American politics, do you think Democrats at the time didn't?