Ban addressed the UN General Assembly as the first speaker during the opening session of its high-level debate involving more than 140 heads of state and government.
The Syrian question is expected to take centre stage at the summit, with high-level meetings including one between US President Barack Obama and Russian President Vladimir Putin later in the day.
The responsibility of ending the Syrian crisis lies with warring parties, Ban said, but the multi-sided conflict was also driven by"regional powers and rivalries."
He called for the situation to be referred to the International Criminal Court, a measure that had been vetoed by Russia last year.
The UN chief also called on all countries to shoulder their responsibilities when it comes to assisting refugees, many of whom are fleeing the conflicts in Syria and Iraq.
"We must combat discrimination - in the 21st century, we should not be building fences or walls," Ban said, referring to Hungary's decision to construct a fence along its southern border to stop the flow of migrants.
Ban also warned that the situation in Yemen was dire, with 80 per cent of the country's population in need of humanitarian assistance and urged an end to airstrikes by a coalition led by Saudi Arabia noting they are the cause of most of the casualties.
"I call for an end to bombings, which are also destroying Yemeni cities, infrastructure and heritage," he said.
He called on UN member states to contribute to the 20-billion-dollar humanitarian appeal that is needed this year to help the 100 million people worldwide who are in need of immediate humanitarian assistance.