The nation's banks will be helped by the recent plunge in interest rates, the investment banking firm Salomon Brothers said today. But the deep recession will cause an increase in their problem loans, as some companies will find it difficult to pay on time.

Because the costs of obtaining funds are falling more swiftly than the interest rates banks are charging their corporate customers, bank earnings will grow faster than earlier predicted, according to Warren R. Marcus, head of Salomon's bank securities department.

Marcus predicted that bank earnings would grow between 9 percent and 12 percent this year. While that is below the giant-sized rises of 18 percent last year and 27 percent in 1978, the outlook is much better than that of a "a couple of months ago," Marcus told reporters.

"In March, I would have guessed bank earnings would have risen 5 to 6 percent this year," he said.

In the last two months, short-term interest rates have fallen sharply as the severe recession has dried up demand for money and credit. Interest rates banks pay for much of the money they borrow for re-lending to customers have fallen by 50 percent.

At the same time, because loans outstanding have fallen as well, banks have to "purchase" fewer funds on the open market than they did in March. The bank prime rate, which was 20 percent in early April, is 13 percent at most banks, although Citibank, New York's biggest, lowered its prime rate to 12.5 percent today.

Marcus, the principal author of Salomon's annual study of the 35 biggest U.S. banks, said that the spread between what banks pay for their funds and what they lend those funds for should widen from about 2.89 percent in 1979 to 3 or 3.05 percent this year.

But if the decline in short-term rates augurs well, for banks, the deteriorating economy poses problems.

As the recession affects profits and straps companies for cash, the number fo outstanding loans that are not being repaid will rise.

Marcus said that the increase in problem loans probably will not become apparent until 1981. "These things happen with a lag," he said.

Nevertheless, Marcus said the steady reduction in problem loans that banks have achieved since 1976 will come to an end in 1980.

In 1976, as a result of the severe economic recession of 1974 and 1975, the banks that Salomon Brothers studies had $14.3 billion of "non-performing" loans on their books, about 2.7 percent of their outstanding loans.

By last year, those banks had reduced their problem loans to about one percent of total assets. Some banks were much higher -- First Chicago, for example, had 2.8 percent of its loans classified as "non-performing" -- while others were much lower than the average. Morgan Guaranty Trust had only 0.3 percent of its assets characterized as "non-performing."

Marcus said that he does not think there will be "the degree of credit problems we saw" in the aftermath of the 1974-1975 recession.

"I'm convinced the bad experiences of 1974 and 1975 have made a real impact on bank lending postures. There are none of the excesses that surfaced in 1974 and 1975, especially in the real estate area," he said.

He said that bankers were surprised by the credit problems in 1974, in large part because the nation had not gone through such a trying period since the 1930s.

"Problems that could have been minimalized weren't dealt with early in 1974. Now banks are on top of their loan portfolios," Marcus said.