Maybe the solution is to enforce the immigration regs and penalize employers who hire undocumented people.

That would be the most effective way, which is why they don't do it. They need that exploitation to make record profits for themselves and the shareholders.

Supreme court needs to overturn Ford v Dodge Brothers, the case that established that companies are only accountable to their shareholders, since a century later its clear that companies have an obligation to multiple parties and not just the shareholders. A company needs to comply with safety regulations to protect its employees to reduce the risk of having to pay for medical treatment as a result of an injury, paying employees better ensures a higher level of productivity in most cases which itself could be a benefit to the shareholders, companies need to make sure their costumers are happy so they need to offer a quality product that won't poison them, etc. Plenty of examples of a company having responsibilities to others, in some cases we can argue this practice itself benefits the shareholders and yet we still see corporations believing this antiquated idea that was decided by an ancient court case from an entirely different era.

Then providing some protections for all workers, even the illegal ones, while also passing legislation that prohibits and punishes out-sourcing. Prior to Reagan most undocumented workers were just Mexicans who crossed the border to do seasonal work or a daily routine (maids) for localities near the border, then they'd go home at the end of the day or season. These jobs aren't really competitive so let them do these things so long as they aren't hurting people. H1B's definitely steal good paying jobs, thats why we hear about Indian's scamming their way into them with fake degrees and certifications. Highly competitive jobs shouldn't be up for grabs like that.

All these call centers shipped to India that access are personal info is going to bite us in the ass sooner or later. Make it illegal for companies to do that and we'll see jobs pour back into the USA. No reason not to, a large and stable middle class fuels economic growth.