The prime lending rate climbed to 13 1/2 percent at many of the nation's major banks today, as the Federal Reserve Board continues to keep pressure on interest rates in an attempt to fight inflation and boost the sagging dollar on international money exchanges.

The quarter-point increase logged today puts the prime rate at a new record high. Each time banks have raised the rate since Aug. 28 -- when Chase Manhattan breached the then-reigning 12 percent milestone -- a new record rate has been reached.

The rise had been anticipated by most analysts, many of whom now think short-term interest rates are at or near their peak after a dizzying two and a half months of steady increases.

In other developments, the Commerce Department said its index of leading economic indicators was flat in August while the Agriculture Department said farm prices rose 1.5 percent in September.

The Commerce Department index, which is supposed to foreshadow economic developments, has been signalling a weak economy since last April; the nation's total output of goods and services did decline in the second quarter.

However, early and incomplete figures from the Commerce Department suggest the economy may rebound slightly during the current quarter, which ends Sunday. Nonetheless, the so-called real gross national product still will be lower than it was during the first three months of the year.

In another interest-rate action today, the Department of Housing and Urban Development said government-backed loans for apartment buildings will go to a record 10 percent on Monday, up from 9 1/2 percent for all FHA multi-family projects.

Many economists think the nation already is in the middle of a recession, although they are divided over whether to anticipate a serious economic decline or a mild one.

The index of leading indicators will not help resolve that debate. A spokesman said the index confirms that the economy is sluggish, but tells nothing about the magnitude of the decline. Four of the 10 indicators available behaved favorably in August, while six declined.

While the economy was getting inconclusive news from the Commerce Department index, it got bad news on the inflation front from the Agriculture Department. The 1.5 percent rise in farm prices follows a 4 percent decline in August.

A moderation in food price increases in recent months had helped offset the steep climb in energy prices, although the inflation rate remained in the double-digit range.

Now, just as the economy has about fnished absorbing the higher oil prices put into effect last spring by petroleum-producing countries, it is hit by another round of food price increases.

The Agriculture Department said that higher meat, milk and wheat prices were responsible for the increases. Both hog and cattle prices rose. Farm prices are 11 percent higher than they were a year ago.

Citibank, the nation's second largest, announced this morning that it would boost its prime rate from 13 1/4 percent to 13 1/2 percent. Most major banks followed Citibank's lead.

The prime rate supposedly is the interest a bank charges its best corporate customers for a short-term loan, although many loans are made above or below the prevailing prime rate. Despite the sharp rise in interest rates in recent months, loan demand has remained strong.

Business and industrial loans, which have grown $3.5 billion since midyear, declined $342 million at major New York banks this week. The money supply -- currency in circulation and checking accounts -- also declined slightly after weeks of heft increases.

If loan demand should level off or decline and the money supply should slow its growth, then the Fed would be able to ease the steady pressure it had placed on interest rates since mid-August. The Federal Reserve tightens monetary policy to fight inflation by slowing down borrowing and reducing money growth.

However, analysts said the continued weakness of the dollar on foreign exchange markets may force the Federal Reserve to keep interest rates high.