It is clear that the children of foreign diplomats are not guaranteed automatic citizenship by the Citizenship Clause. It is also clear that the children of permanent legal residents are guaranteed automatic citizenship by the Citizenship Clause.
What is not clear is whether the children of illegal immigrants have such a cons utional right. Congress has long extended automatic citizenship to them as a matter of law, so we have not had the occasion for the courts to decide the issue.
According to wikipedia, this is what the US Attorney General had to say 5 years after the adoption of the Citizenship Clause: "The word 'jurisdiction' must be understood to mean absolute and complete jurisdiction, such as the United States had over its citizens before the adoption of this amendment. Aliens, among whom are persons born here and naturalized abroad, dwelling or being in this country, are subject to the jurisdiction of the United States only to a limited extent. Political and military rights and duties do not pertain to them."
Whatever the answer is, I would prefer Congress to take a crack at it, since there already is arguable justification to limit birthright citizenship. I'd rather put of cons utional amendment until it's absolutely necessary.