The number of passengers riding Metro subway trains climbed last month to nearly 314,000 a day, setting a record for the 7-year-old rail system and apparently signaling a reversal of a ridership decline that had troubled transit officials.

"We finally have turned it around," said Robert A. Pickett, assistant planning director for the transit authority. He attributed the upturn partly to recent improvements in the subway system that have apparently made service more reliable and attracted more riders.

The transit authority reported yesterday that the number of rail trips taken by passengers on an average weekday in June rose to 313,780--a substantial increase over the 303,074 average for June of last year. Last month's weekday average exceeded by more than 3,000 rides the previous record of 310,011 trips set in June 1980 during a period of nationwide gasoline shortages.

Pickett said last month's increase appeared especially significant because it followed a rise in subway fares in April. Higher fares normally lead to decreases in ridership, and Metro officials had forecast a drop of 11,000 rail passengers a day after the April fare boost.

"In spite of the fare increase, the ridership is up," Pickett said. In addition, he described the June ridership average as encouraging because it appeared to confirm an upward trend that had begun to emerge in recent months.

Last year, Metro officials had expressed dismay as the number of subway passengers failed to increase as rapidly as had been expected. In April of last year, subway ridership had fallen off slightly, marking the first decrease since the rail system opened.

The discouraging trend last year was attributed by Metro officials to cheaper gasoline, high unemployment, a fare rise and less dependable service with more trains breaking down at rush hour. In an attempt to reverse the downturn, the transit authority embarked on an effort to make service more reliable, partly by providing spare trains for use when normal service was disrupted.

Pickett said several other factors apparently contributed to the ridership increase in June, including the recent opening of Metro's Yellow Line, slightly higher gas prices and improvements in the region's economy.

In another development yesterday, Metro officials mounted a lobbying campaign to persuade Congress to appropriate more money for construction of new subway lines. They charged that a Senate move this week to hold down spending would have a "devastating" effect on the Metro system.

The Senate Appropriations Committee voted yesterday to allocate $230 million to the Metro system for subway construction. The amount was the same as the Reagan administration had recommended Jan. 31, but was $45 million less than the House had approved last month.

Metro officials charged yesterday that the Senate proposal would result in delays in completing the Northern Virginia rail extension to Vienna and in building new subway stations in the District and suburban Maryland. An Urban Mass Transportation Administration official disputed Metro's assertions, saying that additional funds were available from other sources.