Open immigration (OI) is the position supported by most ARI-associated Objectivists such as Yaron Brook, Elan Journo, Amy Peikoff and Onkar Ghate.**
According to OI anyone should be able to move from country A to country B (and reside there) so long as he: (1) does not have a criminal record; (2) doesn't have an infectious disease; and (3) isn't a suspected terrorist or terrorist sympathizer.
OI Objectivists will often say that this is a different issue from citizenship (and voting).
A few problems:
1. With respect to voting, in the USA we have birth right citizenship (if you are born here you are a citizen). The children of immigrants will automatically get citizenship and the right to vote. So to the extent that OI Objectivists acknowledge the possibility of immigrants voting leftist (which they occasionally do) the problem is only deferred for a generation. Hispanic citizens for example vote 2 to 1 Democratic.
2. Almost everything government does can be reversed. However, the demographic and cultural changes from OI cannot be reversed.
3. OI is in fact more of a political system (like Democracy or Communism) and not just an immigration policy. With OI, a country will get the political system imposed by the immigrants and there descendants. Now an OI advocate might say "well at least will have open immigration," but there is no guarantee that the immigrants will vote to retain OI. When Israel becomes Islamic as a result of OI (which it certainly will) it is very likely that the new Islamic government will end OI.
4. Getting back to voting, even if a country doesn't have birth right citizenship, sooner or later the immigrants and their children and their children's children will get the right to vote. You can't disenfranchise a group of people permanently. There is no way to stop Muslims from voting when they become the majority in Europe. Just look at the case of Rhodesia and South Africa. (My point here is not to defend these countries but just illustrate a general example.)
An Argument For OI Debunked:
One argument is that OI isn't a problem in a free society because, in a free society, there is freedom of association. Hence you don't have to associate with an immigrant (or immigrant group) if you don't want to. However, we don't have a free society. All sorts of laws compel you to associate with people (in particular housing and employment). So in a sense mass immigration is forced association. In any event we are light years from repealing such laws. And if you can't convince native born that you should have the right to not bake a cake for a wedding what chance is there of convincing tens of millions of third worlders who come from countries with no history of respecting property rights?
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* Harry Binswanger takes a more extreme position. There should be no screening and no border checkpoint. Travel and residency in the US should be no different than moving from Connecticut to New York.
** Leonard Peikoff's position isn't entirely clear.
