The libertarian anarchist community is somewhat split on borders, so let me list all the arguments I've heard for both sides to see which amounts to more.
Pro-border #1: Taxes.¶
Immigrants will be disproportionately the recipients of government welfare, which means taxes will likely be raised and open borders will result in more theft.
A good point. More immigration probably will result in more theft.
Pro-border #2: Voter corruption.¶
Being disproportionately the recipients of welfare, immigrants are likely to vote for more government welfare. They're also likely to be attracted to the democatic party, and therein be poisoned by the rest of the false promises of government.
A more interesting concern, if not more significant. Immigration may cause more people to vote for more oppression.
Pro-border #3: Forced association.¶
Anti-discrimination laws threaten to commit additional acts of force against Americans, who will in some cases not be allowed to exercise freedom of association with the immigrants.
Related to pro-border #1, Nullus Maximus raised this here. But is this so significant? In how many cases will this happen, and how severe will the injustices be? It does exist, but surely this is the weakest point.
Anti-border #1: The borders are illegitimate.¶
Obviously, the state's borders do not enclose private property by any conception of libertarian ethics, so preventing people from crossing them is prima facie an illegitimate use of force.
This is very large; by itself, it seems to preclude any libertarian stance in favor of borders outside of consequentialism. Some libertarians try to interpret the NAP in such a way that closed borders count as self-defense because of the violations of our rights that will occur if they're allowed in, even though the immigrants will not be the ones initiating force or even necessarily asking for it; but the implications of this are questionable at best: what would happen if we applied the same principle to people already in America? Would it mean that Americans who receive lots of welfare are legitimate targets for self-defense? I find it dubious that any self-claimed libertarian would stand by such a thing.
Anti-border #2: Taxes.¶
The wall and border patrol must be built and maintained via theft.
Directly countering pro-border #1. It's likely that this even outweighs pro-border #1 because closed borders don't even stop all immigration; only some of it.
Anti-border #3: They're victims too.¶
People who want to immigrate illegally to America are currently oppressed by other states. They are victims of coercion who are entitled to restitution, so giving tax-funded welfare to them undoes as much injustice as it commits by taxing us.
This was hinted at in Jessica Flanigan's post Political Authority and the Basic Income; it's not welfare that's evil, it's coercion.
Anti-border #4: Violence.¶
Even if we believe it's right to keep immigrants out to defend our property, that doesn't entail that it's right to use the violent means necessary to do so. State-enforced borders will mean border police using violence those who try to enter illegally, and justifying keeping them out is not the same as justifying what harm will come to them at the hands of American police.
Considering all this, I find it quite clear that libertarians must support open borders. The points against borders clearly outweigh those for.