Much of the way we think about immigration today is shaped by the experience of Irish coming here in the mid-19th century and Jews at the turn of the 20th century. These two immigrants streams are the only major ones that bear any resemblance to the poem's "tired," "poor" and "huddled masses" fleeing for their lives, never to look back. For sure, the Irish and Ashkenazim have had an outsized influence on America, if no other reason than they helped shape the character of the newly industrializing cities of the East Coast, but their numbers are a small share of our nation's historical immigration flow. And yet too many open-borders folks see in every one of today's immigrants someone fleeing the Potato Famine or being chased out of Anatevka by the Cossacks.
This doesn't necessarily translate to a specific policy position. You could, for instance, take from this insight the lesson that our immigration policy needs to be more like that of the Persian Gulf states, where large numbers of foreigners come and go to work but aren't incorporated into our society (I've argued against that view here and elsewhere). Or you could conclude that we've outgrown the mass-immigration phase of our nation's development and it's time to move on. But in either case, we need to disenthrall ourselves from the mythology of the Mother of Exiles.
I've written about the Statue of Liberty here, saying
"This whole thing is very strange. Imagine a hypothetical foreign country with immigration problems explaining its policy this way: "We used to have sensible immigration laws, but someone built this damn statue." You'd think they were mad."