Many of the nation's biggest banks yesterday cut their prime lending rate from 15 1/2 to 15 percent, the lowest that key business lending rate has been since November, 1980, and the third Monday in a row it has been trimmed by a half-point.

Analysts said that further reductions in the prime rate are likely over the next several weeks. This should help thousands of ailing businesses, large and small, that have been deeply hurt by high interest rates and simultaneous difficulties in raising prices or increasing sales in the midst of the severe recession.

Even if the rate fell to 14 percent soon, a level many analysts think is possible, the prime would be high by any historical standard.

The stress businesses are feeling was evidenced yesterday in a Commerce Department report that new orders for manufactured goods fell 0.3 percent in June to $156.76 billion seasonally adjusted. New orders presage business income, so a decline in orders indicates further declines in sales for many companies.

Particularly hard hit were orders for durable goods, such as appliances and automobiles, which are important to assembly-line employment. New orders for durables declined 1.9 percent in June to $74.66 billion, the department said, following a 1 percent decline in May.

The June decline in overall factory orders followed a 1 percent increase in May, a figure revised downward yesterday from the earlier reported 1.5 percent increase. April's order figure was also revised, to a 1 percent decline from a previously reported 2.9 percent decline.

Commerce Secretary Malcolm Baldrige said in a statement that the June figures "reflected continued weakness in the industrial sector of the economy." But he remained optimistic that a recovery is imminent. "I believe demand will strengthen in the third quarter, led by consumer spending," he said.

The Commerce report also said that factory shipments in June rose less than 0.1 percent to $160.8 billion after rising 2.9 percent in May. Inventories contracted 0.5 percent in June after rising 2.9 percent in May.

In a separate report, Commerce said construction expenditures of $230.7 billion in June were up 1.3 percent over May levels but still 3 percent below last year. Adjusted for inflation, construction spending was down 7 percent from a year ago, the department said.

The poor showings for factory orders and construction spending reflect businesses' continued caution in the face of a stagnant economy. Last month, the Commerce Department said in a report that it expected the nation's nonagricultural businesses to increase spending on new plant and equipment by only 2 percent this year, a change from the department's prediction three months earlier of a 7.3 percent rise in capital spending in 1982.

Businesses also seem to have slowed their borrowing recently. After strong demand for business loans for several weeks, the total of commercial and business loans at the nation's leading banks was down $1.26 billion in the week ended July 21, the Federal Reserve reported last week, after dipping $789 million a week before. If that trend continues, more downward pressure on the prime rate is likely, analysts say.

Among the major banks that cut their prime lending rate yesterday were Citibank, Chase Manhattan, Manufacturers Hanover Trust, Morgan Guaranty Trust, Chemical Bank, and First National of Chicago. Riggs National Bank, Washington's biggest, also cut its prime rate to 15 percent.

The banks are responding to the recent sharp downward movement in so-called open-market interest rates, where banks and other institutions buy and sell funds. The Federal Reserve Board, architect of monetary policy, has blessed the recent decline by lowering its discount rate from 12 to 11 1/2 percent and then last Friday to 11 percent.

After the Fed announced its discount rate cut late Friday afternoon, interest rates tumbled sharply, with short-term rates falling about three-eighths of a percentage point. The price of long-term government bonds rose as much as $10 for each $1,000 of face value.

The interest rate on three-month certificates of deposit, the big million-dollar securities banks rely on to raise much of their short-term funds, fell from more than 15 percent on June 24 to less than 11.5 percent in recent days.

The recent sharp declines in interest rates are attributable mainly to an easier monetary policy on the part of the Federal Reserve. The Fed mainly pursues its monetary policy by buying and selling government securities in the open market. The discount rate, the interest the Fed charges member banks and other institutions to borrow, usually is used to confirm a policy direction, as it was last Friday.

Fed officials acknowledge that because the money supply seems to be growing within a range they consider tolerable, the central bank can take actions that have the effect of lowering interest rates.