This statement is still true to some degree, but it is nowhere near as true as it was when Emma Lazarus wrote the words you cite in “The New Colossus” in 1883.
In those days, immigration to the United States was essentially free. The Chinese had been excluded by an act of Congress, but there were essentially no restrictions on other immigration. This means that the US was truly welcoming almost all of the “tired” and the “poor” from other countries. Almost anyone who could make it to the US was welcome. There were typically lots of jobs for everyone, even if those jobs were very difficult jobs.
Today, the US still welcomes many immigrants. However, with our country having essentially filled up its territory (no more sparsely populated frontier areas), we do not have the same capacity to accept immigrants. Therefore, we no longer allow unlimited immigration. This has cut back on the ability of the “tired” and the “poor” to come to the US legally.
The US is still a country that accepts immigrants and allows them access to the American dream. However, it is less welcoming than it was 130 years ago.