Er, sort of.
What you should say is the Statue of Liberty and the "Give me your sick, tired, poor ..." meant let people come to the US and
make themselves, not rely on others.
An "open border"policy only has issues when ...
- Productive "Migrant Workers": American businesses abuse them with Visa programs that don't let the productive ones become citizens, let alone allow them to pay well below market value -- especially when there are people taking part of the paycheck on both sides of the Atlantic, and ...
- Unproductive "Undocumented Residents": Are given services and, worse yet, commit fraud in various states, including even forged voter IDs (in states that merely use a DL, which they don't check), as well as the "hidden costs," like their rate of car accidents they have without insurance, etc...
Again, the idea was the immigrant would be given an opportunity to make themselves. I still believe in that, and we have to get back to that ideal.
Don't blame the immigrant for the door mat we give them (unproductive) or abuse we take them for (productive).
Until the left and right both re-focus on letting immigrants make themselves, and get away from either "free votes" (for "free services") and "indentured servitude" (in endless Visa variations), we'll never solve the problem.
Ironically both Sanders and Trump do understand this. Clinton and Bush do not.