President Reagan's budget proposal includes a record $361 million federal payment for the District of Columbia for fiscal 1983, a 7 percent increase over the current level that will help the city meet the soaring costs of Medicaid for the poor and salaries and pension benefits for city employes.

But the increase was more than offset by a new proposal to force the city to assume a substantially greater share of the cost of treating the mentally ill at the federally operated St. Elizabeths Hospital--a policy that threatens to throw Mayor Marion Barry's own 1983 budget proposal out of balance.

"There is no way we can cover such a significant amount at this time without serious jeopardy to our current service levels or steep increases in taxes," said D.C. Budget Director Gladys W. Mack.

The president's new budget proposal offered mixed news for some of Washington's best-known institutions and sites.

Howard University, for instance, would receive $17.3 million more in federal aid for program improvements and to help cover mounting salary costs, and the National Park Service would spend $1.7 million to improve retaining walls in Rock Creek Park.

But federal operating subsidies for the John F. Kennedy Center for the Performing Arts would be cut by $217,000 under Reagan's budget, while the Folger Theatre and the Corcoran Gallery of Art each would lose $288,000 in federal grants.

In addition, Metro officials said that Reagan's proposed budget for mass transit would not provide enough funds to enable Metro to continue construction work here next year, although the officials said they would lobby on Capitol Hill for additional money story on D1 .

The budget proposal to reduce federal assistance to St. Elizabeths, a 126-year-old institution in Southeast Washington that treats many of the city's mentally ill residents, is likely to stir the most controversy.

The administration is asking that the city double its contribution to running the hospital from $23 million this year to nearly $50 million in 1983. But Barry's 1983 budget proposal, now before the City Council, contains only $23 million for St. Elizabeths.

Reagan's budget proposal also would require the city to assume the full cost of treating D.C. residents at the hospital within 10 years. Washington residents comprise 90 percent of the 2,000 inpatients at St. Elizabeths, where operating costs will total an estimated $128 million next year. The hospital also has 3,000 outpatients now.

White House budget officials have argued that the federal government should no longer pay for what in effect is a local hospital facility. However, Department of Health and Human Services officials opposed the OMB plan to precipitously end federal involvement in St. Elizabeths.

The administration's decision to gradually increase the city's share of the federal hospital's costs--a proposal that requires congressional approval--is far less drastic than an earlier plan, offered by the Office of Management and Budget (OMB), that effectively would have meant shutting down the institution.

Moreover, federal officials said the city could offset part of its increased expenditures for the hospital by seeking reimbursements through Medicaid and other government and private hospital insurance plans.

Mack said yesterday that the federal Department of Health and Human Services devised the proposal without consulting D.C. officials. Federal and D.C. officials have talked in the past about creating a quasi-governmental corporation to administer the hospital and jointly share in the cost.

"There's just no way we can absorb such a significant cost without federal assistance," Mack said. "We think it's unfortunate the proposal has been put forward without District input."

Despite the controversy over St. Elizabeths, city officials apparently were pleased with Reagan's proposal for a $361 million federal payment--$25 million more than approved for fiscal 1982.

The annual federal payment, which represents about 20 percent of the city's total operating budget, is granted by Congress in lieu of taxes on federal property within the District and to reimburse the city for services provided to the federal government.

Mayor Barry has boasted that the federal payments have increased dramatically since he took office in 1979, claiming that he has improved the city's relations with the White House and Capitol Hill.

The payments have increased from $250 million in 1979 to $276.5 million in 1980 to $300 million in 1981 to $336.6 million in the current fiscal year.

Barry and his budget officials have counted heavily on an additional increase in the federal payment next year to offset the spiraling medical costs for low-income families and to cover increase benefits for city employes.

Of the proposed $25 million increase in the federal payment, $15 million would be used to cover additional Medicaid costs, $5 million would go for salary increases, and $4.3 million will be paid into the city employes' retirement fund, according to budget director Mack.

Many of the proposed cuts for other area institutions are included in the Interior Department's budget.

Federal aid to help operate and maintain the Kennedy Center would be reduced from $4.1 million this year to $3.8 million in fiscal 1983. The National Park Service currently pays about 76 percent of the center's operating costs, according to an Interior Department budget analyst, but that would be reduced to 70 percent under Reagan's plan.

Reagan also wants to eliminate Park Service grants to the Folger Theatre Group, 201 East Capitol St., and to the Corcoran Gallery, at 17th Street and New York Avenue. Congress first authorized the two $288,000 grants two years ago.

The president has recommended eliminating federal funding for the Interstate Commission on the Potomac River Basin, which was created by four states and the District of Columbia to reduce water pollution and to improve water and land management. This year's budget contains $55,000 for those purposes.

The budget contains funds to begin construction of a Museum of African Art and a Center for Eastern Art, on a site off Independence Avenue near the original Smithsonian Institution Building.

The budget also contains a slight increase in planning and construction funds for the National Zoo--from $1.15 million this year to $1.55 million in fiscal 1983.

The proposed increase in funding for Howard University, a private, predominantly black institution, will eliminate accreditation deficiences by improving the library and several programs, including health science, business and public administration, communications and the graduate school of arts and sciences.

The budget also would provide federal grants to Gallaudet College, a school for the deaf located at Seventh Street and Florida Avenue NE. The proposed funding would provide instruction for about 4,200 students. It also would provide public service programs for about 36,000 people.