Article 31 of the UN refugee convention:
https://en.wikisource.org/wiki/1951_Refugee_ConventionWhat this says is essentially that only countries bordering directly on war zones have an obligation to tolerate the illegal entry of refugees; as soon as a refugee has entered the first so-called "safe third country", his refugee status no longer gives him a free pass when he tries to move on to greener pastures. In particular, those entering illegally from a safe third country can be prosecuted (in full accordance with the refugee convention) for their illegal entry. This, in turn, implies that they have broken their obligations under article 2 of the convention, namely that "Every refugee has duties to the country in which he finds himself, which require in particular that he conform to its laws and regulations [...]".
Since the United States are surrounded by countries in which the clauses of article 1 ("being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion") are not met, they have no obligation under international law to take in any refugees from Venezuela. Neither do Mexico or any of the other Central American countries.
I disagree, and so does the UN.