Initial claims for regular state unemployment benefits totaled 578,000 for the week ended May 15, down 6,000 from the previous week but still a high level, the Labor Department reported yesterday.

The number of persons receiving benefits, however, rose by 56,000 to a level of 4,045,000 in the week ended May 8, reversing declines in the two previous weeks, the department said. Both series of numbers are seasonally adjusted.

Separately, the department also reported a sharp slowing of wage and salary increases, as well as in other compensation paid by employers.

The department's employment cost index for private non-farm workers rose only 7.8 percent in the 12 months ended in March, compared to 9.8 percent for the 12 months of last year and 10.7 percent for the 12 months ended in March, 1981.

Part of the slowdown was attributable to the fact that employers' costs for Social Security payroll taxes did not rise as rapidly this year as they did last year. But the size of wage and salary increases is diminishing rapidly, too.

In the year ended in March, 1981, wages and salaries for private non-farm workers rose 9.3 percent, according to the index. By December, that 12-month rate of change was down to 8.8 percent. By March of this year it had fallen to 8.1 percent as the deepening recession, rising unemployment and slowing inflation combined to hold down workers' pay gains.

The slower rise in compensation affected both union and non-union workers. Union workers' compensation rose 11.8 percent in the year ended in March, 1981, and dropped to 9.1 percent in the following 12 months. For non-union workers, the decline was from 10.2 percent to 7 percent.

The drop was greater for workers in metropolitan areas, where the figure fell to 7.7 percent from 10.9 percent the year before, than for those in other areas, where it went from 10.4 percent to 8.3 percent.