If history goes as it did in OTL, may we assume that other nations do have their OTL immigration restrictions? This is rather important, and becomes more so as we come closer to the present. Certain OTL factors influencing certain forms of migration have been very astutely described in this thread, but one thing is (I feel) missing: the rather obvious fact that migrants tend to go to where they can go. Lots of people migrate because they want a better life, and in many cases, any number of more wealthy countries look as if they might provide a chance to get that. If most countries have restrictions, many migrants will go to those countries that have fewer or no restrictions.
This becomes rather more crucial as we get to the 20th century. It has been noted in this thread that most migration restrictions came about as extensions (or re-implementations) of World War One-era wartime restrictions. That is true. But there is an underlying factor at play here. Keep in mind that during wars in the 19th century, restrictions were also put in place in certain places at certain times, but these were typically lifted after hostilities ended. This changed in the early 20th century. The reason? Well... the 19th century was over, and the first elements of (what would later become) the welfare state were taking shape in many developed Western countries (such as Britain, Germany, France, Belgium, the Netherlands, and of course the USA). At the end of the 19th century, anti-monopoly laws came about. State-organised pension schemes began to emerge. Labour laws became a thing. Stuff like that. The standard of living in the developed Western nations began to rise dramatically.
So if you're a migrant... do you go to country X, which does not have those benefits, or to county Y, which does? Spoiler: it's country Y, ten out of ten times. And that's only natural. The only way for country Y to prevent this (provided it wants to prevent it) is to restrict immigration somehow. Most such developed countries have done this, because they have found that utterly unfettered migration + a welfare state = not really all that fuctional. Many European countries were rather lax about migration in the 19th century, because the government was nothing but a night watchman anyway, and almost no one got any state-funded benefits. But that began to change at the end of the 19th century, and became a completely different situation throughout the 20th. There are various views concerning the costs and benefits of migration. Most people do recognise, however, that totally free migration and a welfare state will not work. Because either you apply the welfare benefits to all newcomers... or you don't.
- If you do, you'd better pray that the majority of them will be net contributors to the treasury, because otherwise, your funds will run out and your system will go bankrupt. Most people, even those who generally favour tolerant migration policies, agree that totally free migration will draw in a truly vast number of unskilled migrants, the large majority of which will not be net contributors to the treasury. There will not be enough jobs for all these newcomers, so... they end up on welfare. (To avoid a flamewar: I stress that this will in most cases be against their will! They will want to work, but will not be able to find any.) So including them in the welfare system will drain the treasury on rather short notice. The rich country receiving all these migrants, more likely than not, will end up getting poorer. A lot poorer, even. If this policy is perpetuated indefinitely, I predict economic collapse.
- If you don't apply the welfare scheme to migrants (or perhaps apply it only to migrants who have become net contributors first, or some other selective method like that) you will most likely still get an enormous number of unskilled migrants, simply because wealthy countries offer more opportunities. (Mexican illegal immigrants to the USA in OTL do not come to get benefits, after all, because they are not eligible for those; they come in the hope of a better life than the one they can hope for in Mexico.) These migrants will be faced with a very difficult situation. Imagine that you are an unskilled migrant in a country with a rather high minimum wage and other such laws, and there are a lot of autochtonous people and a lot of better-skilled migrants there... the chances of finding an above-board job are negligable. Employers will have a lot of candidates to choose from, and they'll only choose the best candidates. So the country ends up with an underclass of unskilled migrants with no prospects. That's not as economically disastrous as the other scenario, but still pretty terrible for a lot of people.
And all these factors are put into overdrive in this particular AH scenario, which is that things go basically as in OTL... except the USA has no immigration restrictions at all. For the reasons outlined above, such restrictions (in one form or another) are pretty much unavoidable for a country that is becoming a welfare state. So other developed Western countries will have such restrictions. And the USA does not. And migrants will go (as I pointed out) where they can go. And particularly the unskilled migrants with few prospects will be the ones refused elsewhere. Those are the ones who will come to the USA in droves. So if the USA doesn't have any restrictions, that leaves two options:
- The USA begins to become a welfare state as in OTL, and due to it having no restrictions while all other developed nations begin to implement them, it becomes the nation that pretty much every desperate immigrant in the world is drawn to. Those who are skilled have a much better chance of getting into other countries, so the USA will largely get the masses that other countries do not want. If it attempts to still develop a welfare state, this will end very poorly. (In actual practice, this scenario is very unlikely, because the implementation of immigration restrictions is a very obvious step, and will be taken.)
- Or the USA doesn't attempt to develop a welfare state at all, and retains free immigration because there is no danger of vast numbers of migrants overextending the system. Nativists will still hate immigrants (mostly because they are competition for low-skill jobs), but all in all, it will work much as it did in the 19th century. (This is really the only way I see this happening.)