The president has threatened to unleash "fire and fury" on North Korea and warned that he may be forced to "totally destroy" the rogue state.
Pyongyang's foreign minister in September stated that Trump had "declared war" on his country and that Kim's regime would consider shooting down American bombers. The White House later described the notion that the U.S. had declared war "absurd."
But Kang warned that "we need to be very careful about military options," adding that they "are there to give strength to diplomacy."
"This is a country that grew out of the total devastation of the Korean War, and over a matter of six, seven decades we've become a thriving democracy ... and a thriving market economy," the foreign minister said. "The idea that another war could wipe that out is just unimaginable."
A key U.S. ally, South Korea is home to more than 50 million people. Some 28,000 American troops are stationed in the country. Seoul, its capital, is just 30 miles from the North Korean border and in range of thousands of artillery pieces that are trained on the city.
North and South Korea are technically still in a state of conflict since the Korean War ended in an armistice, rather than a peace treaty, in 1953. In the decades since the Koreas were partitioned, the North has become an impoverished, militarized pariah, while the South has embraced technology, capitalism and deep ties with the Western world.
The Pentagon has warned that the only way to locate and destroy with complete certainty all components of North Korea's nuclear weapons program is through a ground invasion.
There are about 18,000 civil defense shelters across South Korea. While they would offer protection against North Korea's conventional weapons, they are not designed to withstand a nuclear or chemical attack.
A report by the Congressional Research Service released last month estimates that as many as 25 million people on either side of the border, including more than 100,000 U.S. citizens, could be affected by an escalation of a military conflict on the Korean Peninsula.
However, Kang highlighted a month-and-a-half lull in "provocations," suggesting that the "message is getting through" to North Korea.
In Japan earlier Monday, Trump called Pyongyang’s nuclear ambitions and weapons tests a “threat to the civilized world and international peace and stability.”
"The era of strategic patience is over,” he told journalists in Tokyo alongside Prime Minister Shinzo Abe.
Trump has previously said that Secretary of State Rex Tillerson was "wasting his time" trying to negotiate with Kim, belittling previous negotiation efforts and adding that "only one thing will work."
Kang said Trump's comments on Kim in recent months were an "indication of his strong, strong desire to come to terms with this, to resolve this once and for all."
And while the foreign minister acknowledged that some of Trump's rhetoric had raised anxieties in South Korea, she stressed that her government was focused on the president's "overall tone."
North Korea has said in public statements that it wants an official end to the Korean War. It also wants nothing short of full normalization of relations with the U.S. and to be treated with respect and as an equal in the global arena.
F. Brinley Bruton
F. Brinley Bruton is senior editor in charge of NBC News Digital’s London bureau.