Major commercial banks and the Federal Reserve Board boosted key interest rates another notch yesterday in the face of rising inflation.

Many banks raised their prime lending rate another quarter percentage point to 9 percent, while the Federal Reserve voted over the wishes of its chairman, G. William Miller, to increase the cost of funds to its member banks a similar amount to 7.25 percent.

The Fed's vote was 3 to 2, with Miller in the minority. Observers said it was the first time in memory the Fed had acted to boost its discount rate over the wishes of its chairman. Miller declined comment on his vote.

The increases came as, separately, Treasury Secretary W. Michael Blumenthal warned that rising interest rates pose a danger to the economy. [Details, Page D8.]

A further warning signal came from the Commerce Department, which reported that new factory orders in May increased only 9.7 of a percent, the smallest gain in four months. This is considered another indication of slower economic growth in the months ahead.

One Fed source said the vote by Miller was "political," an attempt to assuage Carter administration fears.

But at the same time. Miller's honeymoon with Wall Street seems to be ending. Analysts increasingly express concern that the Fed, under Miller, has allowed the nation's basic money supply to grow too rapidly. They fear this is fanning the inflationary flames even higher.

"There is a growing credibility gap between the statement of the chairman and the behaviour of the monetary aggregates," commented Lawrence Kudlow, chief monetary analyst for Paine Webber, Jackson & Curtis. "It's the old story of hearing the Fed chairman give strong talk but conduct an expansionist monetary policy."

The basic money supply, known as M-1 has grown at an 8.4 percent rate over the last year. Kudlow said, a rate of growth equaled only once - during 1973, for a short period.

"As a result, inflationary expectations are taking another turn for the worse," he added.

The increase in the prime rate, which banks charge large commercial borrowers, was the fifth this year. The last came only two weeks ago.

Yesterday's hike was initiated by Citibank, the second largest bank in the country, which sets its rate on a formula, and was quickly followed by dozens of other major banks.

The banks attributed the rise to a continuing climb in their own cost of funds and to strong loan demanded from customers.

The last time the prime stood at 9 percent was in January 1975, on the way down from the record peak of 12 percent in 1974, when the Fed slammed down hard on the credit rate and contributed to the deep 1975 recession.

Money market analysts say the behavior of monetary policy and interest rates in the current cycle more and more resembles the 1973-74 period, and say the record rates of that period may be threatened before a downturn arrives.

"The cutting edge of monetary restraint, at least so far, is not clearly visible in the credit market, even though interest rates have increased sharply since early 1977." Henry Kaufman, a partner in Saloman Brothers and the firm's chief economist, told a hearing of the Congressional Joint Economic Committee this week.

But Kaufman said he knew of no "creditworthy borrowers that are shocked" by current interest rates, even though a few years ago they might have been. He said consumers and business executives have factored higher interest rates into their outlook.

The prime rate is the most closely watched of a number of money market interest rates - all of which have been increasing steadily in recent months. While consumer borrowing rates are not tied to the prime, a rise in this rate often presages higher interest costs when consumers go to banks for mortgages or personal loans.

Explaining the latest increase in the discount rate, which was announced after the close of financial markets yesterday afternoon, the Fed said, "The action was taken in recognition of increases that have occurred recently in other short-term interest rates and to bring the discount into closer alignment with short-term rates generally."

Such an increase had been widely anticipated. Banks had been borrowing heavily from the Fed's "discount window" because of the low prevailing rate rather than raise money for loans by more expensive means. And the speculation had been that the boost would be one-half rather than one-quarter percentage point. The two previous increases this year - on Jan. 6 and on May 11 - had each been half a point.

On Thursday, the Fed in its weekly report on money market conditions reported that borrowings from its discount window for the week ending June 21 totaled a hefty $1.58 billion, up from $1.07 billion the week before.