Riggs National Bank, the Washington area's largest financial institution, yesterday reported a 12 percent increase in earnings last year to record levels. Fourth-quarter profits, generally stronger for banks because of rebounding interest rates, jumped 21 percent.

Madison National Bank of Washington also reported strong gains during 1980, while Government Services Savings and Loan Inc. of Bethesda reported a loss ofr the quarter and nine months ended Dec. 31. The S&L's figures reflect the current relatively weak operating posture of thrift institutions.

At Riggs, which owns Central Charge Service, operating profits for the year rose to $24.3 million ($8.13 a share) from $21.8 million ($7.29) in 1979. Fourth-quarter earnings were $5 million compared with $4.1 million; pre-share figures were not available. Net-income figures for the year and quarter were substantially the same as before securities transactions.

Chairman Vincent Burke Jr. said assts rose 17 percent to a record $3.15 billion during the 1980, while deposits increased by 14 1/2 percent to a record $2.5 billion and loan volume expanded by 14 percent to $1.54 billion.

Madison Bank, which like Riggs is planning to establish a holding company this year, reported a 22 percent increase in earning last year to $2.4 million ($5.86 a share) from $1.97 million ($4.80) the previous year. Chairman Donald Menefee said assets increased 10 percent to $174 million, while deposits rose to $156 million from $134 million and loans increased to $104 million from $99 million.

Government Services listed a third-quarter loss of $244,204 compared with earnings of $145,645 (17 cents a share) in the 1979 period. Nine-month results produced a loss of $1.4 million compared with earning of $741,210 (87 cents). Savings deposits declined to $296 million on Sept. 30 but up from $268 million a year earlier.

President A.R.M. Boyle said the third-quarter loss was less than in the previous two quarters. The renewed high-interest environment will penalize thrift industry earnings for the near term, he added.

McCormick & Co., the Baltimore-based spice company, said earnings in the year ended Nov. 30 fell to $14.8 million ($1.31 a share) from $19.4 million ($1.71) as sales rose to $548 million from $457 million. An accounting change reduced the recent year's profits by 32 cents a share.