Trump on Friday put a 120-day hold on allowing refugees into the country, an indefinite ban on refugees from Syria and a 90-day bar on citizens from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
The executive order led to the detention and deportation of hundreds of people arriving at U.S. airports, huge protests in many U.S. cities and a raft of legal challenges amid confusion over its implementation, Reuters reported.
Trump defended the move as vital for U.S. security, but his critics have said his action violated U.S. law and the Constitution.
MSCI's broadest index of Asia-Pacific shares outside Japan fell 0.4 percent in holiday thinned trade.
Australian shares tumbled more than 1 percent, while New Zealand pulled back 0.6 percent.
Japan's Nikkei widened losses to 0.7 percent as demand for the safe-haven yen weighed on exporters.
Pointing to a weaker opening on Wall Street, S&P e-mini futures, Dow futures and Nasdaq futures all fell around 0.3 percent.
"Trump always stated these were policies he would implement. Quite a lot of it was brushed off as 'campaign rhetoric' but he is following through," said James Woods, global investment analyst at Rivkin Securities in Sydney.
"This renews concerns about a trade war with China that would significantly affect both Asian and the global economy," Woods said.
"The biggest threat to markets at the moment is if Trump continues down the path of protectionism without focusing on economic policies."
Several countries including long-standing American allies criticized Trump's directive as discriminatory and divisive, and tens of thousands of people rallied in U.S. cities and at airports on Sundays to express their outrage.
U.S. judges in at least five states blocked federal authorities from enforcing Trump's executive order, but lawyers representing people affected said some authorities were unwilling on Sunday to follow the judges' rulings.
U.S. 10-year Treasury yields were last at 2.4658 percent, down from 2.481 percent as of Friday's close.
The dollar index, which tracks the greenback against a basket of trade-weighted peers, dipped about 0.3 percent to 100.21 in Asian trade.
The dollar also weakened almost 0.7 percent to 114.31 yen on Monday, pulling further away from a one-week high hit Friday.
Adding to pressure on markets, data on Friday showed U.S. economic growth slowed more than expected in the fourth quarter, with GDP rising at a 1.9 percent annual rate, below the 2.2 percent rise expected by economists and the 3.5 percent growth pace logged in the third quarter.
Earnings disappointments also weighed, led by Chevron, whose quarterly profit missed expectations, and Starbucks, which trimmed its full-year revenue forecast.
But the impact across Asia may be delayed, with China, Hong Kong, Taiwan, South Korea, Singapore and Malaysia shut on Monday for the long Lunar New Year holidays.
While U.S. policies are causing some nervousness, investors in Asia will also focus on the Bank of Japan's policy meeting on Tuesday as well as manufacturing and services activity surveys out of China on Wednesday.
Markets will also be watching U.S. manufacturing data and the Federal Reserve meeting's outcome on Thursday, and Friday's non-farm payrolls figure.
In commodities, oil started the week on a negative note, extending declines on signs on growing output in the U.S. that looks set to offset supply cuts by the Organization of Petroleum Exporting Countries and other producers.
U.S. crude retreated 0.3 percent to $52.99 a barrel, adding to Friday's 1.1 percent slide.
Global benchmark Brent crude dropped 0.3 percent to $55.34, after losing 1.3 percent on Friday.
Gold shone amid the pullback in risk markets. Spot gold added more than 0.2 percent to $1,194.25 an ounce.