Hong Kong was also being weighed by another weekend of violence, fuelling worries about possible Chinese intervention in the financial hub, while the unrest has also hit property firms and Macau's casinos.
Washington's latest levies on imports from China took effect on Sunday and were followed later by Beijing's retaliation.
The measures are the latest in the long-running trade war between the world's top two economies, which has rattled global markets and hitting growth across the planet.
Still, Trump said negotiators would meet this month to discuss the issue. "We are talking to China, the meeting is still on," he told reporters.
However, analysts warned there was unlikely to be any end in the near term.
"There is a long way to go," said Shane Oliver, head of investment strategy at AMP Capital Investors. "Share markets may still have to fall further to pressure Trump to resolve the issue."
Tokyo ended the morning session 0.2 percent lower, Sydney and Singapore each shed 0.6 percent, Taipei was off 0.1 percent and Seoul was marginally lower.
But Shanghai rose 0.7 percent after a better-than-expected reading on Chinese factory activity, though investors remain uncertain about the outlook as the trade war bites deeper.
- Violence grips Hong Kong --
Hong Kong sank after a weekend that saw some of the worst violence since protests began three months ago, with the airport targeted again, and demonstrators have called for a general strike on Monday.
The unrest has dragged a range of sectors, with tourist numbers tumbling, hitting casinos and hotel chains, while real estate shares are also being sold off.
"Markets are fretting on the increased likelihood of direct Chinese intervention and what that would mean for the future of one of Asia's leading financial centres," said Jeffrey Halley, senior market analyst for Asia-Pacific at OANDA.
"The answer is, not good, to put it bluntly. The economic impact will surely show in Hong Kong data going forward and may temper the mood of equity traders in Asia as the new month begins."
Oil prices extended Friday's steep losses owing to worries about the impact of the trade war on demand, while dealers were also concerned about reports that the Russian output cut last month fell short of an agreement with OPEC.
"A fissure is forming in OPEC+ compliance, which saw oil prices crater," said Stephen Innes, APAC Market Strategist AxiTrader.
"On the surface, while it is not likely a significant divergence, it's the messaging that Russia is sending that spooks markets. While it could be little more than a tempest in an oil can at this stage, it's worth monitoring nonetheless."