President Reagan's plan to "foster the infrastructure of democracy" abroad is picking up steam on Capitol Hill, thanks to bipartisan cooperation and the efforts of Rep. Dante B. Fascell (D-Fla.), who has worn many hats in the process.

The centerpiece of the effort to promote democracy abroad is a $31.3 million-a-year proposal in the State Department's fiscal 1984 authorization bill. It would create a National Endowment for Democracy to distribute the funds to organs of the Democratic and Republican parties, the AFL-CIO, the U.S. Chamber of Commerce and other institutions in a manner "consistent with broad U.S. national interests."

Details of the plan, which envisions a considerable amount of foreign travel, were devised by a commission called the Democracy Program, whose executive board included Fascell.

Fascell also:

* Introduced the legislation to put it into effect.

* Served as chairman of the Foreign Affairs subcommittee that reported it out.

* Testified on its behalf before the Senate Foreign Relations Committee.

* Will serve as floor manager of the authorization bill that includes the "National Endowment for Democracy Act" among its provisions.

Would become the first chairman of the endowment's 15-member board if the legislation is enacted.

In addition, one of Fascell's aides, R. Spencer Oliver, has signed up as an officer of the newly established National Democratic Institute for International Affairs. That organization is in line for an infusion of $5 million a year for at least two years "to encourage free and democratic institutions throughout the world" by such means as visitor exchange programs.

The institute was incorporated April 28 along with the National Republican Institute for International Affairs in a cooperative spirit reflected by the fact that the nearly identical articles of incorporation were typed on the same typewriter.

The leading members of the boards of the two institutes are Democratic National Committee Chairman Charles T. Manatt and Republican National Committee Chairman Frank Fahrenkopf.

Oliver is also staff director and general counsel of the Commission on Security and Cooperation in Europe, a predominantly congressional agency set up to try to monitor Soviet compliance with the 1975 Helsinki accords. Fascell is the commission's co-chairman.

"The tracks were greased for this all the way," said one Hill veteran who asked not to be identified. "You've got both political parties, labor and business behind it. This is a sweetheart of a sweetheart deal. One of the fears is that it could be turned into a slush fund for travel. That's clearly what the money can be used for."

Fascell could not be reached for comment. Oliver's office said he was "out of the country."

In testimony April 27 before the Senate Foreign Relations Committee, Fascell said he had been concerned for years "with the apparent inability of this country to create effective political, non-military responses to the challenge which our adversaries pose in the competition for ideas throughout the world."

Meanwhile, he said, the Soviets have "tried to penetrate the institutions" of other countries--from political parties to labor movements to educational systems--with "an effective political apparatus."

"Now, as a result of the endowment proposal and the institutions which it would fund," Fascell said, "finally this country has the private sector concept it needs to engage effectively, responsibly, and in a long-range manner in the global competition of ideas."

The CIA used to fund covertly much of what the endowment plan envisions, such as the publication of books and articles "consistent" with democratic ideals. But many of these were halted in 1967 following public disclosure of the CIA's activities.

Under the bill, the National Endowment for Democracy would get $31.3 million a year from the U.S. Information Agency's salaries and expenses account.

The endowment, in turn, would be required to give $13.8 million to the AFL-CIO's existing Free Trade Institute, $5 million each to the Democratic and Republican institutes, and $2.5 million to a foundation established by the U.S. Chamber of Commerce. This would leave $5 million that the endowment, which would not carry out any programs itself, would disburse to other "private sector" grantees.

The bill itself also names Fascell as "interim chairman of the endowment." Other members of the endowment's board would include Manatt, Fahrenkopf, AFL-CIO President Lane Kirkland, and Senate Foreign Relations Committee Chairman Charles H. Percy (R-Ill.).

But Fascell, and presumably Percy, would have to step down before the endowment started disbursing any funds, aides say, because of a statute prohibiting members of Congress from serving on any board that disburses executive branch funds.

In Senate testimony, Fascell said the legislation would include "a solid and continuous process of congressional oversight to insure that the institution itself and those it funds fulfill the purposes for which Congress has authorized those funds." He added that "all of the endowment-funded programs, obviously, will be conducted openly by their private sector sponsors."

Those sponsors, however, also would be entitled to take in private funds, Fascell subcommittee staff director Virginia Schlundt said.