Neglecting racial differences, immigration is just population growth. If immigration (or letting women work) made wages go down, wages would have become 0 a long time ago, since having babies would also make wages go down. Whatever it was when there was 1 person, it would now be 1/x of that wage, which would be basically 0.
Let's talk about an economy with exactly one person in it.
There is one guy. He makes one iphone a year. Using only his labour somehow, and neglecting profit and stuff since it's a superfluous complication. It costs $100.*
This guy gets a wife. Let's say she also gets a job making the iphone. Supply of labour doubles,** so price of labour halves. (Not really halves, but close enough.) Oh, but labour was the only input. Yes, he's paid only $50. But iphones now only cost $50. The supply of iphones also doubles, which means the price also halves.*** His wages go down and cost of living goes down in lock step.
In the real world this cancelling effect means the prices don't go anywhere. They're both paid $100. This means instead of $100 demanding iphones, there's $200 demanding iphones. The demand for iphones doubles with the supply of iphones, which means the price goes nowhere. She's paid $100 and buys his iphone, which pays him $100 so he buys her iphone.
Likewise in a real economy, you can do fantastically complicated calculations and find out that, yes, increasing the supply of labour increases the supply of product at the same time, or equivalently that increasing labour also increases demand, so demand and supply changes cancel out.
In the completely real world Mexican immigration actually makes Americans richer in the short term.**** Mexicans typically send their paycheque back to Mexico, meaning the supply stays in America but the demand goes back to Mexico. Meaning America has more raw stuff. Which means America is richer.
In the long term the increased demand in Mexico will cause imports from America, however. Exactly like the pressure of a gas dropping and sucking in neighbouring gas.
*(I guess he sells it to himself since nobody else has money, since nobody else exists.)
**(If she is really exactly as productive as he is, he's a loser and they're going to divorce when she clues into this, but never mind. Spherical cows.)
***(More precisely, whatever price factor you get for doubling supply will be exactly the same for both iphones and labour.)
****(Neglecting labour supply shocks. The economy can react only so fast, because people are stupid.)
Increasing the labour supply also increases the supply of goods that labour demands. If you copy-paste a country next to itself, they don't both suddenly become poor because the supply of labour doubled. If they could support themselves before they will be able to support themselves after.
Immigration isn't a stupid idea because it's bad for the labour market. It's a stupid idea exactly because it isn't bad for the labour market. Someone who is worthless in Mexico will remain worthless if they move to America. Fixes nothing. If they wanted to fix things they should have imported American managers and businessmen, rather than exporting labour. Improving your country is racist, though: it makes all the countries that don't improve super envious.
Primarily folk think increasing population decreases labour value because they lived under Malthusianism for like 200,000 years and they can't overcome their base instincts. As with any good, increasing the supply of food has decreasing marginal returns. Each calorie costs a little bit more than the previous calorie, meaning, if you increase population, the supply of stuff left over after having eaten is less per person. America is nowhere near its carrying capacity so this is wholly irrelevant in present times. Enjoy it while it lasts.