For the sake of argument, let's take as a given that high net migration is undesirable for cultural reasons and, therefore, cutting net migration is desirable.
In order to avoid a significant resulting drop in living standards (and absent some radical solutuion that no government of any developed nation seems close to finding) you need to avoid a shrinking and aging populace. That means raising birth rates, significantly.
This is where right wing or centrist parties run into problems. A few generations ago, it was possible to sustain a large family off a single income. These days, its a struggle for even relatively high earning, dual income households to afford a relatively small family. One of the main drivers of this is insufficiently restrained capitalism, which has seen a huge, prolonged and increasing transfer of wealth from the working and middle classes to the very richest. The mega rich are getting richer, everyone else is getting poorer.
If you want to reduce immigration, you need to create conditions that allow and encourage people to have more children which, in all likelihood, requires wealth to flow back from the mega rich to the rest of us. Right wing parties though, and particularly the likes of Trump, want to either preserve the status quo or further accelerate it's problems.
Stripping away abortion access and limiting access to contraception are other ways of forcing the birthrate upwards, and the US right do that a bit, but they are generally and understandably unpopular policies.
Funnily enough, some on the right appear to be waking up to this issue - see, for example, Suella Braverman's recent(ish) comments for scrapping the child tax benefit cap to be scrapped but, fundamentally, right wing economic policy is incompatible with the notion of a meaningful redistribution of wealth.
The right successfully electioneer off of immigration related concerns but not only do they fail to propose solutions for these issues but, usually, they make them even worse.
Mainstream left wing parties aren't all that much better; they seldom offer radical solutions because, in a world where the vast majority of mainstream media is owned and shaped by the mega rich (and absorbed by the voters) that renders them unelectable.
The above probably sounds like I'm advocating for communism. Yes, the birth rate issue might be improved in a communist system but they bring a whole set of their own problems.
Were I involved in left wing politics, I wouldn't wish to embrace demonizing immigrants but I would acknowledge that it is a concern that many people have and explain to those people (many of whom, are economically left leaning anyway) that the only acceptable and sustainable way to decrease net immigration is through tackling wealth inequality and tinkering around the edges wont cut it. That isn't an easy ask in a global economy but it likely needs to be the direction in which we move.
Right wing populists offer no solutions but they do articulate many people's concerns (and, indeed, fuel them). Even if left wing people are less inclined to object to high immigration, it is an issue they could leverage better to advocate for a more economically just society.