Lady Liberty did not welcome immigrants who were sneaking into the US, but instead who were legally coming to America and who were processed at Castle Garden and Ellis Island. Not everyone got into the US and many were turned away for various reasons. Immigrants faced Ellis Island with trepidation worrying that they would not be found "worthy" of entering the US. They was no welfare in those days and immigrants knew they were on their own and had to make their own way.
This is far different from those who are coming knowing that they will be taken care of by the US government. Would you send your kids to cross Mexico in the hands of the "coyotes" if you knew that if they got into the US there would be no one to take care of them, no free medical care, no shelters, no schools etc. and they would just be living on the streets?
