In an immigration system, there are essentially two points of enforcement: the border and employers.
The current system focuses on the border, as eight million undocumented migrants have managed to get over the border and find employment. Conservatives want a crack-down on employers. However, industries like agriculture, food processing, construction and hospitality services depend intrinsically on migrant labor, and given a choice between bankruptcy and hiring undocumented labor, employers will go with the latter every time. By default, therefore, border enforcement is the central focus of Trump administration policy.
A market-based visa (MBV) system, by contrast, relies on employer enforcement. This is feasible if
- The number of visas materially covers market demand (say, +/- 7% of actual), and
- Employers can access an unlimited number of background-checked workers on short notice, subject to a visa price set by the market
Under these conditions, employers--and the migrants themselves--will enforce the system.
Let's demonstrate by analogy. The US does, in fact, have a sector with thousands of jobs openings, good pay and yet minimal black market labor. That sector? Trucking. Today, there are 100,000 open trucker jobs in the US paying $45-75k, but with minimal black market activity. Virtually all the drivers in the US hold commercial driving licenses. Why?
The answer is simple. A driver's license is comparatively easy to get. Many trucking companies would even pay a driver to get one. On the other hand, if an applicant cannot get a license, he is almost certainly not fit to drive a truck. That is obviously an important consideration for, say, FedEx, both because they do not want their trucks in accidents and because they rightly fear liability. Thus, an individual needs a commercial license to drive a truck not because of heavy-handed government enforcement, but because the license actually conveys information to the employer, and because it represents a reputational and liability risk. In that sense, the system is self-enforcing.
If there are enough work visas to cover the market, then the same logic applies to migrant labor. If a migrant who is not a criminal can purchase a visa anytime, anywhere, then the only migrants ineligible for visas are criminals. Employers do not want to hire criminals. Why would an employer do that, when a million non-criminal Mexicans would willingly take the work on a day's notice? And imagine that an employer hired a criminal -- someone he knew could not obtain a work visa (which he would, using a smartphone-based system) -- and that migrant committed a serious crime. That could constitute aiding and abetting, with unlimited legal liability. Under the circumstances, an employer would have a substantial incentive to reject an applicant without a visa, just as FedEx will not hire a driver without an appropriate license.
Nor would legal migrants permit it. A legal migrant who just paid $4,000 for a visa would actively resist their employer hiring people off the books. So in this world, the migrant workers themselves will be a key driver of enforcement. For both these reasons, dangerous criminals will be frozen out of the legal job market.
And this can be achieved while leaving the border essentially open. Back in Mexico, experienced migrants will say, "Well, the US system is terrific. But it's expensive to pay the fee and hang out there if you're not working. So you want to get work lined up as soon as possible, and you'll do really well if you're working ten hours a day, six days a week, because the visa fee is fixed either way. If you're not working, though, it's best to come back to Mexico. It's cheaper anyway, you can see all the jobs on offer and apply over the internet, and you can go back anytime you want. But whatever you do, don't get caught without a visa. Employers will not hire you without one, because there are a million legal migrants lining up to take your job. And if you're caught by the authorities, you are cooked. You won't be able to work again in the US for years. Moreover, if the other migrants find out you don't have a visa, they will inform on both you and the employer to the authorities. They paid good money for their visa, and they expect you do the same. So, just follow the procedures. Go through the bureaucratic background check, and become visa-eligible. After that, work the internet and your circle of friends and family and jump in legally when you get an opening. If you work long, smart and hard, you'll make good money."
That's how enforcement works in a market-based system.