Could a US citizen born through “birth tourism” become President?
It seems the question may have been answered over 120 years ago. Chief Justice Fuller seemed to think so and he didn't much care for the possibility.
United States v. Wong Kim Ark [p715]: FULLER, C.J., Dissenting Opinion, March 28, 1898. In comments concerning the Court's opinion, CJ Fuller was suggesting that the majority opinion concerning "natural-born citizen", by implication, meant that a child of a birth tourist (modern term) could lead to the possibility that child could become president.
By the fifth clause of the first section of article two of the Constitution, it is provided that:
No person except a natural-born citizen, or a citizen of the United States, at the time of the adoption of the Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty-five years, and been fourteen years a resident within the United States.
In the convention, it was, says Mr. Bancroft,
objected that no number of years could properly prepare a foreigner for that place; but as men of other lands had spilled their blood in the cause of the United States, and had assisted at every stage of the formation of their institutions, on the seventh of September, it was unanimously settled that foreign-born residents of fourteen years who should be citizens at the time of the formation of the Constitution are eligible to the office of President.
2 Bancroft Hist. U.S. Const. 193.
Considering the circumstances surrounding the framing of the Constitution, I submit that it is unreasonable to conclude that "natural-born citizen" applied to everybody born within the geographical tract known as the United States, irrespective of circumstances, and that the children of foreigners, happening to be born to them while passing through the country, whether of royal parentage or not, or whether of the Mongolian, Malay or other race, were eligible to the Presidency, while children of our citizens, born abroad, were not.
The Court applied English common law to reach its interpretation of "subject to the jurisdiction thereof" thus limiting its scope to the children of ministers and consuls. CJ Fuller disputes that "subject to the jurisdiction thereof" means only that; but also includes children of those temporarily visiting the United States; reasoning the parents are still loyal to and subject to the jurisdiction of their home country. Since such people are not subject to the jurisdiction of the U.S. then neither are their children who happen to be born in the U.S.
At p728-729, CJ Fuller adds:
The privileges or immunities which, by the second clause of the amendment, the States are forbidden to abridge are the privileges or immunities pertaining to citizenship of the United States, but that clause also places an inhibition on the States from depriving any person of life, liberty or property, and from denying "to any person within its jurisdiction, the equal protection of the laws," that is, of its own laws -- the laws to which its own citizens are subjected.
The jurisdiction of the State is necessarily local, and the limitation relates to rights primarily secured by the States, and not by the United States. Jurisdiction, as applied to the General Government, embraces international relations; as applied to the State, it refers simply to its power over persons and things within its particular limits.
These considerations lead to the conclusion that the rule in respect of citizenship of the United States prior to the Fourteenth Amendment differed from the English common law rule in vital particulars, and, among others, in that it did not recognize allegiance as indelible, and in that it did recognize an essential difference between birth during temporary, and birth during permanent, residence. If children born in the United States were deemed presumptively and generally citizens, this was not so when they were born of aliens whose residence was merely temporary, either in fact or in point of law.
CJ Fuller interpreted the Constitution and prior law in a manner consistent with my understanding as given in my original answer.
[Original answer, minor editing.]
Possibly Currently, but that is based on an interpretation of the Citizenship clause in the 14th Amendment.
All persons born or naturalized in the United States and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
Some future Court could interpret the clause differently.
For example, of the State wherein they reside, taken narrowly, would mean that such a child, not being a lawful resident of any state at time of birth, is not a citizen of any state and, therefore, cannot be a citizen of the United States. (That child's mother would have to have been a resident of a state.)
That, of course, would end birth tourism.
In United States v. Wong Kim Ark, it was the Court's interpretation of "subject to the jurisdiction thereof" and not the text of the 14th Amendment that allowed for "birth tourism". If a new case were to go before the Court, the Court could overturn the opinion in Wong Kim Ark (though after more than 120 years that seems unlikely).
Alternatively, Congress could, under Amendment 14, Section 5, pass a law to define that "subject to the jurisdiction thereof" excludes: children of ministers and consuls; children of those visiting the United States on a visa; children of those who have been granted entry on a request for asylum, until such time as asylum has been granted; children of those who are unlawfully in the United States; etc.
Those provisions would eliminate "birth tourism" and so-called "anchor babies".