Yes, they can, because they already are.
In an earlier note, we talked about the Relocation Wage, the hourly wage Mexicans would have to earn to make it worth their while to come to the US, and we calculated this around $6.50 / hour. If border enforcement is effective, then the realized wage should be higher than this level. If it is not, then we would expect observed wages to come in around the Relocation Wage. Our analysis suggests that the border is largely ineffective and wages have been around the Relocation Wage, due to wage theft, poor utilization, and hassle costs (risk of deportation, inability to operate normally, etc.)
Thus, Mexicans appear to be realizing $6.50 / hour on typical US unskilled wages of $10 / hour, representing an effective tax rate of 35%.
In a market-based system, we look to monetize theses costs in a fee to the government, which is worth about $30 bn net to the Federal budget -- but is achieved at essentially no cost to illegal immigrants as a whole. We are simply replacing predation and dead-weight costs in a black market system with a formal fee in a legalized system.
The range of outcomes, however, is dramatically altered. Today, the Mexicans who lose the border crossing lottery die in the desert, or are raped, kidnapped, trafficked or incarcerated. That would end entirely, as there is no point in entering illegally if a legal channel is always available at a market rate for a volume of visas which materially cover the entire market.
Similarly, those undocumented migrants already in the US would see much reduced wage theft and higher utilization rates (or lower costs, if they leave the US seasonally), and obviously much greater ease of living and working in the US.
The top 10% would see some decline in net income. On the other hand, the more successful have the most to lose from deportation and the greatest opportunity for advancement in a legal system.
I cannot over-emphasize how much this would change the image of Mexicans in the US. There are a number of conservative radio hosts and columnists on this email list, and some of them really want the Mexicans out. I can assure you, though, if a Mexican day laborer is paying $5,000 in visa fees for nothing more than the right to work and basic healthcare coverage, that migrant will earn the respect of even the most hardened immigration foes. Legalization brings normalization. Today, no one much challenges alcohol consumption, although it causes far more damage to the US than immigrants do. We are used to it, and now to gambling and increasingly, to marijuana. The very act of legalization changes public perceptions dramatically.