President Trump has recently said that he plans to issue an executive order denying citizenship to the children of illegal aliens. The Fourteenth Amendment, though, says,
All persons born ... in the United States, and subject to the jurisdiction thereof, are citizens of the United States.
The relevant federal statute (8 U.S.C. § 1401) likewise says,
The following shall be nationals and citizens of the United States at birth ... a person born in the United States, and subject to the jurisdiction thereof.
An executive order can't supersede either the Constitution or a constitutionally valid federal statute, so the premise behind any such order must be that children of illegal aliens aren't covered by the Amendment and § 1401, presumably because they aren't "subject to the jurisdiction" of the United States. And indeed Senator Harry Reid, back in the early Clinton Administation, proposed a bill that would have taken that very view (though also purporting to rely on Congress's "power to enforce, by appropriate legislation, the provisions of" the Fourteenth Amendment):
In the exercise of its powers under section 5 of the Fourteenth ... Amendment ..., the Congress has determined and hereby declares that any person born after the date of enactment of this title to a mother who is neither a citizen of the United States nor admitted to the United States as a lawful permanent resident, and which person is a national or citizen of another country of which either of his or her natural parents is a national or citizen, or is entitled upon application to become a national or citizen of such country, shall be considered as born subject to the jurisdiction of that foreign country and not subject to the jurisdiction of the United States within the meaning of section 1 of such Article and shall therefore not be a citizen of the United States or of any State solely by reason of physical presence within the United States at the moment of birth.
Now I personally think that categorical birthright citizenship is a bad idea; it would be better if children born in the U.S. to illegal aliens, or to legal alien tourists, didn't get U.S. citizenship as a result (though perhaps the answer might be different as to children of legal permanent residents, or to children who have lived here for long enough, or some such). U.S. citizenship is one of the most valuable things in the world, and we generally don't let people get hugely valuable things because of the criminal acts (even if only mildly criminal) of their parents, or for that matter because their parents were lawfully visiting the U.S. at a particular time. Whatever the basis for deciding who our future fellow citizens -- and thus the future rulers of the nation -- should be, the fact that the child was born on our territory shouldn't be enough.
This having been said, the Constitution seems pretty clear to me, even if I disagree with the rule it sets forth: Being the child of illegal aliens (as opposed to, say, the child of foreign diplomats) doesn't stop you from being "subject to the jurisdiction" of the U.S. Jurisdiction is an entity's power to impose its legal will on someone, and the U.S. unquestionably has the power to do that for children of illegal aliens as much as for children of legal aliens or of citizens. People who commit a crime, including the crime of illegal entry, don't somehow elude the jurisdiction of the U.S. as a result. Likewise, the children of people who commit this crime are subject to our jurisdiction as well.
That, of course, is just the short version -- I thought I'd also pass along a longer version, from Jim Ho, who was just appointed to the Fifth Circuit earlier this year. Jim is a very smart guy, as well as a pretty solid conservative; he clerked for Justice Clarence Thomas, and spent time as Chief Counsel to Senator John Cornyn, and was also Solicitor General of Texas (immediately following Ted Cruz). He has actually studied the subject closely; here is an excerpt from a Wall Street Journal op-ed of his from 2011:
In the aftermath of the Civil War, members of the 39th Congress proposed amending the Constitution to reverse the Supreme Court's notorious 1857 Dred Scott v. Sandford ruling denying citizenship to slaves. The result is the first sentence of the 14th Amendment: "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States."
The plain meaning of this language is clear. A foreign national living in the United States is "subject to the jurisdiction thereof" because he is legally required to obey U.S. law. (By contrast, a foreign diplomat who travels here on behalf of a foreign sovereign enjoys diplomatic immunity from — and thus is not subject to the jurisdiction of — U.S. law.)
During congressional debates, both proponents and opponents of the citizenship clause agreed with this interpretation of the 14th Amendment. For example, Pennsylvania Sen. Edgar Cowan opposed the clause precisely because it would extend birthright citizenship to the U.S.-born children of Chinese laborers and other noncitizens who "owe [the U.S.] no allegiance [and] who pretend to owe none."
Tellingly, Cowan's racially charged opposition was met with the following response from California Sen. John Conness: "The proposition before us ... relates simply in that respect to the children begotten of Chinese parents in California, and it is proposed to declare that they shall be citizens.... I am in favor of doing so.... We are entirely ready to accept the provision proposed in this constitutional amendment."
Supreme Court precedent further reinforces this view of the 14th Amendment. In 1898, the court held that a U.S.-born child of Chinese immigrants was entitled to citizenship. In United States v. Wong Kim Ark, it held that the "14th Amendment affirms the ancient and fundamental rule of citizenship by birth within the territory ... including all children here born of resident aliens."
The court reiterated this view in Plyler v. Doe (1982). The majority held — and the dissent agreed — that the 14th Amendment extends to anyone "who is subject to the laws of a state," including the U.S.-born children of illegal aliens. Likewise, in INS v. Rios-Pineda (1985), the court again unanimously agreed that a child born to an undocumented immigrant was in fact a U.S. citizen.
You can also read his longer (10-page) analysis of the question, published in 2006, here; and you can hear a debate between Ho and Chapman University Prof. John Eastman here. You can also see a view contrary to Ho's, and supportive of Congressional power (though not Presidential power) to resolve the question here, from Yale Prof. Peter Schuck and Penn Prof. Rogers Smith.