Stock prices started off slowly this morning, but finished strong in the last three hours of trading.

The Dow Jones Industrial Average of 30 stocks, probably the most closely followed indicator of stock behavior, climbed 9.39 points today, to close at 964.08. That is the highest the Dow average has been since March 17, 1977 when it hit 964.84.

While stock prices were continuing their 16-week climb, bond prices were sagging. Corporate bonds were down sharply in light trading and Treasury issues were off about $10 on a face value of $1,000.

The bond market is being hit not only by renewed fears of higher interest rates, but also by a big surge in corporate bond offerings that is depressing bond prices.

Companies, that had borrowed heavily from their banks and the commericial paper market when interest rates were at record levels last spring, are now trying to pay off their short-term borrowings with the long-term money they are raising in the bond market.

Corporations floated record levels of new bonds in May and June and near-record amounts in July. The bond markets are anticipating another huge round of bond offereings this week.

Because bond prices move in the opposite direction from interest rates, the jump in short-term interest rates in the last several weeks has hurt as well.

Some analysts think the Fed is tightening its monetary policy, while others think it is on the same course it has taken for months but had to take steps to keep money growth in check.

The Federal Open Market Committee, the central bankhs policy group, meets Tuesday in Washington to consider what course money policy should take for the next month.

But if the bond markets are in the dodldrums, the stock markets are in heaven. Since April 16, the Dow average has climbed more than 200 points.

Analysts expect the market climb to end soon, as all booms must, but each time stock prices stall, a new round of buying begins. Institutions dominated today's trading, and one analyst said there were more big "block" trades.

Trading volume on the New York Stock Exchange was strong, at 44.75 million shares, but lighter than on Friday, when 59.18 million shares changed hands.

The New York Stock Exchange's composite index climbed 0.64 point to an all-time high of 71.45. On the New York Stock Exchange, 964 stocks rose in price on the day while 594 fell.

On the American Stock Exchange, 380 stocks advanced while 249 declined. The Amex index rose 2.84 to 322.25.

Computer stocks were among the strongest performers today, as they were last week.

Petroleum stocks have bounced around in recent weeks as inventors weighed recent increases in oil company earnings against a growing glut of oil in the world and recent price cuts by some major oil companies. For the most part, oil stocks did well today.