It's a common view among members of his party, including Sen. Marco Rubio (R-Fla.), a rival for the presidential nod.
Yet some economists and demographers who have studied Mexican immigration suggest that stricter security at the border could actually increase the number of undocumented immigrants in the country. One group of researchers estimates that by 2010, increased border enforcement over the decades had increased the number of unauthorized migrants in this country by 44 percent.
Crossing the border has become more difficult, but not difficult enough to prevent many Mexican families from settling here. Yet if the border were less heavily policed, some experts argue, many migrants would be crossing it illegally, traveling between their homes and families in Mexico and temporary jobs here. Given security at the border, some of these itinerant workers have decided to simplify things by staying here permanently, the reasoning goes.
If so, more undocumented immigrants are residing in the United States as a result of border enforcement, often bringing their families along with them.
Republicans' focus on border security is "simply misplaced and counterproductive and far as I can tell serves no constructive purpose whatsoever, but that’s where the debate is," argues Wayne Cornelius, a political scientist at the University of California, San Diego.
He describes the increase in the undocumented population as a massive "unintended consequence" of border enforcement on the huge market for immigrant labor in the Southwest. It's a striking critique of a party that has long warned of the dangers of government intervention in the economy.
Limiting immigration from Mexico, though, has been a goal that Democrats and Republicans have shared for most of the half century since 1964. That was when Congress voted to end a guest-worker agreement with Mexico that had allowed migrant laborers into the United States since the Second World War. The workers kept on coming illegally, despite Congress.
And that was the beginning of undocumented immigration. Leonard F. Chapman, the Marine Corps general and President Nixon's immigration commissioner, called it "a growing, silent invasion of illegal aliens" in 1976. He was among the first to argue for stricter security at the border.
Many of these immigrants came to settle, but many others only stayed a few months before returning, according to Douglas Massey, a sociologist at Princeton University. He and his colleagues began conducting surveys of Mexican families on both sides of the border in 1982.
They've collected detailed information about who is crossing the border over the decades, asking Mexicans questions about their families, their jobs, their financial circumstances, whether they crossed legally or illegally, when and where they crossed, and whether and how much they paid a guide, one of the smugglers known at the border as coyotes.
The researchers found that in 1986, about three in five of those who crossed the border had returned to Mexico within 12 months, as shown in this chart.
That same year, Congress passed the first major law increasing security at the border with Mexico. Since then, the United States has dedicated vast resources -- drones, cameras, manpower and hundreds of miles of fencing -- to securing the border. All of that has cost tens of billions of dollars.
Yet these additional security measures haven't prevented many immigrants from entering the country across the Southern border, research suggests. To be sure, making the crossing has become riskier. Just about everyone who crosses illegally now hires a coyote. And the coyotes are charging more than ever, at more than $2,700.
Coyotes can charge high fees, because they almost always find ways around, under or through whatever is in their way. For decades, an immigrant who tried to enter the country illegally was almost guaranteed to find a way, even if he or she was detained by the Border Patrol on the first crossing.
Between 1970 and 1988, the vast majority of undocumented immigrants crossed either at San Diego or at El Paso. When federal agents cracked down -- in El Paso in 1993 and in San Diego the following year -- immigrants simply began hiking through Arizona's Sonoran Desert, Massey's survey data show.
"This was a Bill Clinton project," recalls Cornelius, who has separately been conducting his own surveys of migrants since 1975.
His data show the same trends that Massey and his colleagues have identified. Cornelius says that in recent years, to get around the 651 miles of fence built along the border under President George W. Bush and President Obama, coyotes are going through the front door, concealing their clients in vehicles or presenting false papers in order to smuggle them past customs agents at official ports of entry.
In the past few years, the rate of successful crossings has declined a little, but most still make it through eventually, according to data.
Massey's data as well as federal statistics suggest that the number of Mexicans attempting to enter the United States without papers has declined sharply since 2000. At this point, as many undocumented immigrants are returning home as are entering illegally, and there are now fewer undocumented immigrants living here now than ever.
Yet the data suggest that decline is not a result of increased enforcement at the border. Undocumented immigration has declined because the Mexican economy has expanded while the U.S. economy has not done well, and because Mexico's population is getting older. Young adults are the most likely to immigrate.
Despite the stiff prices the coyotes were charging, it was still worth it for many undocumented Mexican migrants to cross into the United States as long as they had a hope a finding a good job here. Doing it more than once, though, may not be worth it anymore. Massey and others argue that's the more important consequence of the increase in the costs -- and the dangers -- of crossing illegally.
Migrant laborers who might have come for a few months in the past whenever their families needed money and then returned to Mexico are now more likely to stay, the data show -- bringing their families with them, or starting new families here.
Mark Krikorian, executive director of the conservative Center for Immigration Studies, rejects this version of the history of the Mexican border. He argues that the reason that fewer immigrants are returning is not that they are trying to avoid the expense of coming back, but that the seasonal, rural economy of migrant agricultural labor is being replaced by a resident workforce in permanent occupations in major cities, such as construction and child care.
Krikorian dismisses as "laughable" the idea that border enforcement has increased the number of undocumented immigrants living in the country.
"It's completely malarkey," he said. "That's just a fantasy, really an ideologically driven fantasy."
For their parts, the proponents of that idea agree that other factors would have steadily increased the undocumented population in the United States in any case -- the shift to permanent employment, the boom in Mexico's population, and the economic advantages that young Mexicans could gain by moving to the United States.
Proponents do argue that some immigrants have responded to the hardening of the border by staying put north of it, rather than south of it. For that reason, they say, border security has exacerbated the underlying demographic and economic trends driving Mexicans into this country.
In an analysis of their data, Massey and his colleagues found that while there was no statistically significant correlation between the Border Patrol's budget on undocumented migration, the budget was highly correlated with whether undocumented immigrants returned to Mexico.
In an unpublished estimate, Massey and his colleagues calculated that the population of undocumented immigrants in the United States in 2010 -- roughly 14 million, by his estimate -- was 4.3 million greater than it would have been had Congress not increased the Border Patrol's budget above its level in 1986. In other words, Massey argues, border security has increased the undocumented population by 44 percent.
"We'd have a much smaller undocumented population now if we’d done absolutely nothing," Massey said.
Some doubt that the increase in the undocumented population due to border enforcement was so drastic, and they argue that in any case, Congress could constrain undocumented immigration.
If policymakers wanted fewer immigrants to enter the country illegally, they should have taken a different approach, said Pia Orrenius, an economist at the Federal Reserve Bank of Dallas.
She noted that away from the border, federal agents have more effective means of deterring immigration that can reduce the undocumented population. Strictly enforcing the laws against undocumented immigrants working makes it harder for them to get by. The result is that illegally crossing the border for work is less attractive to would-be migrants in Mexico, and those who are here have less reason to stay.
"This really hurts people in the pocketbook every day," Orrenius said.
Historically, though, Congress has spent taxpayers' money on security at the border, rather than on enforcement in the interior. As noted in one report from the nonpartisan Migration Policy Institute that critiqued congressional priorities, spending on border enforcement has exceeded spending on all other immigration programs combined. Customs and Border Patrol received $12.4 billion in 2014.
Some of the GOP candidates have been talking about stricter enforcement in the interior, including Rubio and Scott Walker. "Part of doing this is put the onus on employers, getting them E-Verify and tools to do that," the Wisconsin governor told CNN's Chris Wallace earlier this year, referring to the federal government's electronic program for verifying employees' eligibility to work.
"Border enforcement is the sexy option," Cornelius said. "Politicians can go to the border and do stand-ups."