The nation's unemployment rate was unchanged in August at a post-World War II high of 9.8 percent, with 10.8 million Americans out of work, the Labor Department reported yesterday.

The unchanged rate is a possible sign that the worst of the recession may be over. But several analysts cautioned yesterday that the jobless rate could climb this month, and that the economy has not yet begun to improve.

"If you look at the details" of yesterday's official report -- particularly the drop in factory jobs and a 0.3-hour decline in the factory workweek -- "it is discouraging indeed," said Donald Straszheim of Wharton Econometrics.

There also has been a sudden increase in the number of initial claims for state unemployment benefits since the middle of August, which could translate into a higher jobless total this month, analysts said.

White House spokesman Larry Speakes said President Reagan regards the unemployment rate "as still at an unacceptably high level." But he said "nearly all the other economic news is good."

In the District, the city government reported yesterday that unemployment for July remained virtually unchanged at a record 11.4 percent, while the rate for the overall metropolitan area was 6.3 percent. Details, Page B1

Meanwhile, apparent fears of a crisis in the world financial system led to a rush for "quality" in hectic stock market trading yesterday.

The Dow Jones Industrial Average climbed 15.73 points as 130.9 million shares changed hands on the New York Stock Exchange. It was the third-busiest trading day in exchange history.

Gold prices soared $20 an ounce to close in New York at $458, the highest level in almost a year.

The Federal Reserve Board, in the meantime, yesterday reported a $1.5 billion increase in the nation's money supply in the latest week. The increase was at the upper end of market expectations. With the money supply growing fairly rapidly for two successive weeks, the Fed is unlikely to encourage further declines in interest rates just yet, analysts said.

Commerce Secretary Malcolm Baldrige, reacting to the unemployment report, said yesterday "there are signs that the situation in the weakest sectors will improve in the months ahead." He said "the household survey showed a slight increase in the total number of jobs" and that those in the work force "have experienced a real gain in purchasing power compared with a year ago." This rise was 0.7 percent, the Labor Department reported.

Administration officials hope that the dramatic decline in interest rates in recent weeks will boost the economy off the floor of recession.

Details of the August employment report showed that black unemployment climbed further, with 18.8 percent of the black labor force out of work, compared with 18.5 percent in July. Unemployment among black teen-agers rose from 49.7 percent to 51.6 percent.

The rate of unemployment among adult men edged up 0.1 percent to a postwar high of 8.9 percent.

Although the Labor Department survey showed a slight rise in employment to 99.8 million, businesses reported a 210,000 drop in jobs in August. Over half of this decline came in manufacturing industry.

Economists have been surprised by the persistence of the recession.

"There just simply hasn't been any strength evident yet in consumer spending or any of the other . . . categories in the economy that will make it recover," Straszheim said.

Yesterday's unemployment report "suggests that in August probably the economy was still sinking, or at best flat," Otto Eckstein of Data Resources Inc. said. "The tax cut didn't do it," he added. "Let's face it. In the end it turned out that it was not enough to lift the economy . . . . Now, hope has shifted . . . to lower interest rates."

The recession has hit particularly hard at autos, construction and related industries. "The decline in factory jobs occurred almost entirely within the major metals and metal-using industries," said Janet Norwood, commissioner of the Bureau of Labor Statistics. Five industries in this group "have accounted for over half of the 1.9 million overall decline in nonfarm payroll employment since the pre-recession peak in July, 1981," she said.

With the auto industry in the doldrums again, there was a sharp increase in unemployment among auto workers last month after several months of improving job prospects. These workers now face a jobless rate of 20.8 percent, Norwood said. "Since March, 1979, employment in the auto industry has declined by one-third," she added.

The number of people who lost their jobs rose in August, the Labor Department reported. Job losers accounted for 58 percent of the unemployed, and the average duration of unemployment rose to 16.2 weeks.