President Reagan asked representatives of 34 nations yesterday to help show the world how to build democracy by the expansion of free elections.

The "key to democracy" is a free citizen "placing his vote in a ballot box" and thereby affecting his "own destiny," Reagan said on the opening day of a three-day Conference on Free Elections.

Reagan said at a White House luncheon that he hopes this conference on democratic institutions and processes will produce "a positive program for international action" to give new momentum to the spread of freedom.

The current meeting of specialists on democratic processes from nations ranging from Australia to Turkey is an outgrowth of Reagan's speech before the British parliament June 8 urging a "campaign for democracy."

Costa Rican President Luis Alberto Monge Alvarez, responding to Reagan's luncheon toast, said his small, poor country can offer neither military nor economic power to the campaign. But what it can supply, he said, is evidence "to defeat the false theories . . . that say democracy is a plant that cannot grow in the tropics" or "that democracy is a sort of dessert that can only be afforded by rich countries."

In opening the conference at the State Department, Secretary of State George P. Shultz listed these prime objectives: to "provide concrete assistance to countries interested in establishing free elections"; to advocate "the right to free elections more actively"; to expand "human rights discussions and activities to include positive steps that foster free institutions" and to "do more to publicize the success of democracy."

Speakers included Italian Prime Minister Giovanni Spadolini; Alhaji Shehu Musa, special representative of Nigerian President Alhaji Shehu Shagari, and Jeane J. Kirkpatrick, U.S. ambassador to the United Nations.

Today the conference scene shifts to the Senate Caucus Room, where the discussions include a report on Tuesday's national elections.