The American Enterprise Institute, a prestigious local think tank that has served as an intellectual and personnel wellspring of the pro-business Reagan administration, is facing a financial drought.

The institute has cut its annual operating budget by 13 percent, imposed a hiring freeze that has left 20 clerical positions unfilled, and this month suspended production of its regular television and radio show, "Public Policy Forum," that was carried on more than 600 commerical, public and cable stations.

"Frankly, we have some cash flow problems," acknowledged Joseph Brady, AEI vice president for development.

Sources both inside and outside the conservative institute say its fund-raising may fall as much as $2 million short of its current $10 million annual operating budget, but AEI President William J. Baroody Jr. insisted: "I fully expect to exceed the $10 million in revenues."

Some reasons for AEI's problems:

* It receives fully 50 percent of its annual operating budget from corporate contributions, and thus is less cushioned against the impact of a recession than other public policy institutes that have a broader funding base.

* It faces growing competition for the same corporate and foundation dollar from a hungry collection of new entrants into the public policy research field.

* It is, in the view of some, a victim of its own success. More than 20 fellows and scholars left AEI in 1981 to join the Reagan administration, including such luminaries as United Nations Ambassador Jeane J. Kirkpatrick, West German Ambassador Arthur Burns, Federal Trade Commission chairman James Miller III, former chairman of the Council of Economic Advisors Murray Weidenbaum and White House communications director David Gergen.

"You can't lose a Jim Miller and not suffer a bit," said one think-tank official who asked not to be identified. "Jim Miller was the top guy in Washington on regulatory reform issues. Jeane Kirkpatrick is absolutely irreplaceable."

Baroody does not agree that his institute has lost any competitive edge from the ballyhooed and probably temporary loss of these big guns.

"I don't attribute a financial penalty to that"," he said. "We may have lost a Murray Weidenbaum, but we still have a Herb Stein and a Paul McCracken."

Some corporate givers are delaying, and a dozen or so are withholding, their promised contributions this year because they face their own financial dilemmas, he noted. In addition, the institute has simply refrained from soliciting money from some past pillars of support such as International Harvester, which is one the verge of bankruptcy.

And yet Baroody believes that the AEI's corporate gifts will actually increase this year over last. That is because the institute's stable of corporate givers has grown from 225 in 1975 to 650 today.

The AEI is trying to broaden its fundraising among individual donors with a three-year direct mail campaign to gain 50,000 individual donors.

Baroody, who took over the institute in 1977 from his father, has been criticized in some quarters for overpaying the AEI's scholars, of whom there are about 35. Former president Ford, for example, is a distinguished fellow who receives $100,000 annually for visiting a handful of colleges, setting up a yearly seminar for business leaders and attending a few AEI-sponsored social events. "He more than justifies AEI's stipend," said Baroody.

Washington is filling up with new entrants into the public policy research field, such as the Institute of International Economics and the Roosevelt Center for American Policy Studies, and Baroody acknowledges the competition is making things tougher.

But other established think tanks don't seem to be suffering as much as AEI. The conservative Heritage Foundation's operating budget has climbed from $5.3 million in 1980 to a projected $9 million in 1983. It plans to move to newer, larger quarters next year.

And the Brookings Institution, the liberal equivalent of AEI, has not had to resort to austerity cuts. Brookings has an endowment that provides it with about 20 percent of its $12 million budget; government contracts account for 21 percent, sales and fees 25 percent, foundations 15 percent and corporate gifts just 13 percent.