Moon said a second summit between Kim and Trump will prove to be the "most critical moment" for North Korea's denuclearisation.
According to Moon, Trump would be inclined to meet Kim's demands if the North Korean capital, Pyongyang, discards its nuclear arsenal.
The military said no unusual North Korean military activity was reported on Saturday. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply.
The pair "reaffirmed their commitment to achieve the final, fully verified denuclearisation" of North Korea, Mr. Trump's spokeswoman Sarah Sanders said.
"The crisis is still continuing because of North Korea's development of nuclear, missile and weapons program and materials", Ban said.
They agreed on the need for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK understands that denuclearization is the only path", Sanders said, using the North´s official name, the Democratic People´s Republic of Korea.
Mr Trump has frequently described a warm personal relationship with Mr Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded USA presidents since the 1950s.
In recent months, however, the two sides have begun to remove landmines and destroy military bunkers at parts of the border as part of efforts to improve long-strained relations.
The South Koreans and their counterparts will live in the train, inspecting two railway lines for a total of 18 days - one linking the North's southernmost Kaesong City to Sinuiju City near the Chinese border, and the other connecting Mount Kumgang near the inter-Korean border to Tumen River bordering Russian Federation in the east.
"Through the railways that will be connected in one, the South and North will prosper together and peace in the Korean Peninsula will become firmer", South Korean Unification Minister Cho Myoung-gyon said during a ceremony at Dorasan Station near the border. But prospects of a fourth Moon-Kim meeting have recently dimmed, with negotiations on denuclearising the North grinding to a halt.
The North Korean leader said he would visit Seoul "in the near future", during his third summit with Moon in September.
North Korean and South Korean soldiers are pictured during an operation to reconnect a road across the Military Demarcation Line inside the Demilitarised Zone (DMZ) separating the two Koreas (Novemeber 22, 2018).