The current interpretation giving citizenship to children of illegals and tourists is bogus. It isn’t supported by the 14th Amendment at all. It was something made up by bureaucrats and judges in the past 50 years or so.
Sen. Vitter has legislation to end “birthright” citizenship for children born to people who are under the jurisdiction of another country. (Rand Paul supported that legislation in 2011 but hasn’t said anything about it since then; another reason he’s losing voters to Trump.)
And that's the reason why any change must be VERY carefully worded so that some future evil administration (and there will be one if the US survives...) will pervert the lack of protection as to where they can autonomously strip citizenship of whomever they disapprove...and the court would approve of the action since there's nothing preventing it.
Again, I wholeheartedly agree with doing what is necessary to eliminate the concept of anchor babies. But we have to tread carefully so we don't leave our progeny vulnerable to some future tyrant.