Despite a falling unemployment rate, labor costs in the United States continue to rise at a moderate annual rate of 3.4 percent, the Labor Department reported yesterday.

The department's employment cost index, which covers wages, salaries and fringe benefits, increased 1.2 percent in the third quarter and 3.4 percent in the 12 months ended in September, the department said.

Many forecasters and financial market participants, concerned about a possible acceleration of inflation, have been watching various measures of wages to spot any indication that labor costs have begun to increase more rapidly. Unless productivity -- output per hour worked -- also rose, a faster rate of wage change likely would mean higher inflation. Some expected that lower unemployment would raise employment costs by increasing competition to hire workers.

However, the quarterly increase in the index was almost identical to a 1.1 percent rise in the third quarter of last year. The index rose 0.7 percent in the second quarter, but because the figures are not adjusted for seasonal variation, comparing changes from one quarter to the next may be misleading.

For the 12 months ended in June, the increase was 3.3 percent, slightly less than the 3.4 percent in the year ended in September. For the year ended in September 1986, the rise was 3.6 percent.

Within the overall index, there were some divergent movements among industries and regions.

Labor costs in goods-producing industries, many of which face strong competition from imported products, went up 2.6 percent in the past year compared with a 3.1 percent rise the year before. For service-producing industries, the index was little changed, going up 3.8 percent in the past year and 3.7 percent a year earlier.

Within the latter group, changes in employment costs accelerated from a 4.4 percent rise a year ago to 4.8 percent in the year ended in September for services, including a jump from 3.8 percent to 4.3 percent for health services. Employment costs for federal, state and local governments and other public bodies rose less rapidly in the past year, down from a 4.8 percent gain to 4.1 percent, the department said.

While the decline in the civilian unemployment rate, which fell from 7 percent in September 1986, to 5.9 percent last month, does not appear so far to have had a significant impact on the rate of increase in the employment cost index, differences in unemployment have affected regional changes in the index.

In the past 12 months, labor costs again rose most rapidly in the northeast region, which has by far the lowest average unemployment rate among the four regions for which figures are given. There the index rose 4.5 percent, up from 4.2 percent a year earlier.

The index also rose more rapidly in the Midwest, from a 2.5 percent gain to 3.1 percent, and in the West, from 2.3 percent to 2.8 percent. However, in the South, labor costs rose 2.7 percent in the past year, down from 3.3 percent a year earlier.