The area's banking industry continued to show signs of rebounding from a difficult first quarter yesterday as two of the city's largest banks reported steep increases in second-quarter profits, figures that reflect an easing of record-high interest rates.

The city's largest bank, Riggs National Bank, reported a 23.7 percent increase in second-quarter profits compared to the same quarter last year. Riggs said profits for the quarter rose to $7.1 million ($2.37 a share) from 1979 profits for the same quarter of $5.74 million ($1.92).

The Riggs figures come just one day after American Security Bank, the city's second-largest, reported a 30 percent profits jump.

During the first quarter of the year, Riggs had reported only a one-half percent profits increase. For that quarter, Riggs listed profits of $5.4 million. Six-month earnings were $12.5 million ($4.19) vs. $11.1 million ($3.73).

As of June 30, deposits at Riggs rose to an all-time high of more than $2.3 billion, up 9 percent over last year, and assets rose to $2.9 billion, up 11 percent over the figures on the same day in 1979.

National Savings and Trust Co. yesterday reported a 29 percent increase in second-quarter profits compared with the same quarter last year, making the three-month period one of NS&T's most successful quarters ever.

The quarter's earnings were $1.33 million ($2.92 a share), up from $1.03 million ($2.26) for the same quarter of 1979. This gain by the city's fifth-largest commercial bank follow a first quarter that saw bank profits, hurt by high interest rates, dip by 11 percent.

Boosted by second quarter results, however, the bank reported six-month profits of $2.16 million ($4.74), up 9.7 percent over the first six months of 1979.

NS&T also reported a continuing drop in deposits, which as of June 30 were $395 million, compared with $430 million at the same time last year. Since the end of the first quarter of 1980, NS&T lost another $4 million in deposits.