"We continue to strive for the reasonable and not the emotional approach.of the kindergarten fight between children".
The US administration has refused to offer North Korea incentives to open negotiations and has ramped up threats of military action to force leader Kim to change course.
Mr Trump also said China was imposing major banking sanctions, but there was no immediate confirmation from the North's most important trading partner.
Analysts said fresh threats from North Korea added to the jitters.
A limited amount of petroleum allowed under the United Nations resolution will still be exported to North Korea.
"Foreign financial institutions are now on notice that going forward they can choose to do business with the United States or with North Korea, but not both", Mnuchin said.
The quake comes amid soaring tensions over Pyongyang's weapons programme, with the firing of two missiles over Japan in recent weeks and its sixth and largest nuclear test earlier this month.
As with much that has transpired lately in the U.S.
"And if the test doesn't go according to plan, you could have population at risk, too", he said.
She said Kim's aim in developing nuclear weapons is "to fulfill a long-term desire on the part of the North Korean regime to reunify the Korean Peninsula under the Kim family regime and proliferate these weapons and blackmail other countries".
Hours before Kim's statement, Trump announced stiffer new sanctions on North Korea as he met his South Korean and Japanese counterparts in NY. His comments were believed to be the first direct message ever issued by a North Korean leader, Reuters reported.
Kim on Friday threatened the "highest level of hard-line countermeasure in history" in a tirade against Trump's warning that Washington would "totally destroy" the North if the USA or its allies were threatened.
"We have no idea about what actions could be taken as it will be ordered by leader Kim Jong-un".
"We are prepared to use the full range of military capabilities to defend the USA homeland and our allies". It did not name specific weapons systems.
"It could be the most powerful detonation of an H-bomb in the Pacific", Ri said. "Why not?" he said when asked whether there could be talks with Pyongyang.
Defence secretary Jim Mattis hinted that Trump's threat may not be mere bluster when he refused to deny reports that the Pentagon was considering using tactical nuclear weapons against Pyongyang.
North Korean dictator Kim Jong-Un complained in a defiant statement released by North Korea's KNCA state media of Trump's strong statements, saying they "had never been heard from any of his predecessors".