In the currency markets, the dollar index, which measures the greenback against a basket of currencies, traded around its lowest since November.
The dollar index was at 97.657 at 2:28 p.m. HK/SIN, off levels over 98 on Wednesday and down from levels over 99 last week.
The greenback had climbed after the November election, in part on expectations that parts of Trump's proposed policies would be reflationary, which would spur interest rates higher and drive dollar inflows.
The market appeared to be unwinding that expectation.
"Everyone, including us, has proclaimed that the economic fundamentals are so good that the markets can withstand a political crisis. But an impeachment fight is another story – it would be demoralizing, not good for consumer confidence, and an enormous source of uncertainty for investors, who hate uncertainty," Greg Valliere, chief global strategist at Horizon Investments, said in a note on Wednesday. "This is a deadly serious crisis."
Prices of safe-haven play gold gave back some of their overnight gains on rising political risk. Spot gold was down 0.31 percent at $1,256.40 at 2:31 p.m. HK/SIN, after rising nearly 2 percent overnight and touching its highest levels since late April.
"The move has been driven by 'there's never just one cockroach' approach to political uncertainty, rather than a fundamental investment reassessment structurally of gold itself," Jeffrey Halley, senior market analyst at OANDA, said in a note on Thursday. "Whilst it is evident the investor world does not like cockroaches, gold itself will now trade on headlines rather than fundamentals."
In Japan, exporters were mostly lower, weighed by the yen's surge. A stronger yen tends to weaken exporters' overseas earnings when they are repatriated to their home currency.
Shares of Toyota lost 1.72 percent, Sony lost 2.26 percent and Honda was off 2.23 percent.
Australia's market was weighed by continued declines in the heavily weighted banking sector. Westpac tumbled 3.11 percent after the stock went ex-dividend and ANZ lost 0.38 percent.
The sector has been under pressure since the Australian budget, announced last week, included plans to impose a levy on the earnings of large banks.
Bucking the market drop, shares of Fairfax Media ended up 6.90 percent at A$1.24 after Reuters reported it was considering a second surprise takeover bid.
U.S. private equity firm Hellman & Friedman made a takeover bid of as much as A$2.87 billion ($2.13 billion), valuing the company at A$1.225-A$1.25 a share, Reuters reported.
That compared with an earlier rival bid of A$1.20 a share from TPG Capital Management and Ontario Teachers' Pension Plan Board, Reuters reported.
In Hong Kong, shares of Tencent tacked on 1.46 percent to close at HK$236.60 a stock, with Reuters noting that profit of 14.3 billion yuan ($2.08 billion) for the first quarter, up 58 percent on-year, was above its poll forecast for 13.2 billion yuan. Earnings were reported after the market close on Wednesday.
Stocks in the U.S. finished lower following the latest headlines from Washington, with major indexes recording their worst sessions since last September. The Dow Jones industrial average fell 1.78 percent or 372.82 points to finish at 20,606.93, the S&P 500 tumbled 1.82 percent or 43.64 points to close at 2,357.03 and the Nasdaq dropped 2.57 percent or 158.63 to close finish at 6,011.24.