The US President wants $5.7bn to build a steel barrier, which would deliver on his signature campaign pledge.
In the eight-minute address carried live by all the major US television networks, Trump said the federal government remained shut because of the Democrats.
He said of the situation at the border: "This is a humanitarian crisis, a crisis of the heart and a crisis of the soul."
Trump said an as-yet-unratified revamp of the North American Free Trade Agreement would pay for the wall.
The president also said that 90% of heroin sold in the US came from Mexico.
He also pointed out that Democrats have in the past supported a physical barrier.
In 2006, senators Chuck Schumer, Barack Obama, Hillary Clinton and Joe Biden voted in favour of 700 miles (1,120km) of fencing on the nearly 2,000-mile border under the Secure Fence Act.
Trump cited cases of American citizens "savagely murdered in cold blood" by undocumented immigrants.
"How much more American blood will be shed before Congress does its job?" he asked.
“I’ve met with dozens of families whose loved ones were stolen by illegal immigration. I’ve held the hands of the weeping mothers and embraced the grief-stricken fathers. So sad. So terrible,” Trump added.
Trump said his intention to build the physical barrier is not out of hatred for Mexicans, but love for his country men and women.
"People build walls not because they hate those on the outside. They love the people on the inside," he said.
Trump, who will visit the Mexican border in person on Thursday, invited the Democrats to return to the White House to meet with him on Wednesday, saying it was “immoral” for “politicians to do nothing.” He said they could resolve the standoff in “45 minutes.”
In a brief rebuttal, House Speaker Nancy Pelosi and Senate minority leader Chuck Schumer demanded that Trump end the shutdown.
The California congresswoman added: "And the fact is President Trump must stop holding the American people hostage, must stop manufacturing a crisis, and must reopen the government."
Schumer accused Trump of trying to "govern by temper tantrum".
He said Trump “just used the backdrop of the Oval Office to manufacture a crisis, stoke fear and divert attention from the turmoil in his administration.”
"President Trump has appealed to fear, not facts. Division, not unity," the New York senator said.
He concluded: "The symbol of America should be the Statue of Liberty, not a 30-foot wall."
Democrats argue that maintenance of existing border fencing, hi-tech tools to scan vehicles crossing at ports of entry, and hiring more personnel would be cheaper and more effective than a wall.
Democrats, who recently took control of the House of Representatives, are adamantly opposed to giving Trump the funds to build the wall.
The ongoing closure of a quarter of federal agencies is the second-longest in US history, leaving hundreds of thousands of government workers unpaid.
Trump has been discussing the idea of declaring a national emergency to allow him to move forward with the wall without getting congressional approval for the billions he’s requested. But he did not mention that in his address.
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