Declaring that "ours is a country of refugees," President Carter said yesterday that the United States will continue "to provide an open heart and open arms" to those fleeing Cuba and that he will ask Congress for additional money to care for their needs.

In a speech to the League of Women Voters national convention, the president said the plight of the refugees has been greatly aggravated by "the inhumane approach" of Cuban President Fidel Castro.

Carter specifically pledged that the United States will receive, "with understanding as expeditiously as we can," the 387 Cubans who have taken refuge inside the American diplomatic mission in Havana and are hiding from what the president called "mob violence instigated by Castro himself."

The Cubans fled inside the small U.S. mission last Friday after being attacked by about 20 men armed with chains, pipes and clubs. The U.S. officials have blamed the Cuban government for instigating the attack.

Cuba, meanwhile, has demanded that the men inside the mission -- numbering 308 of the 387 -- turn themselves over unconditionally to government authorities. Most of them are former political prisoners.

White House press secretary Jody Powell said after Carter's speech that the administration has not decided on how large a supplemental appropriation to seek from Congress to deal with the flood of new Cuban refugees. Last week, the president authorized spending up to $10 million from the refugee emergency fund to establish a special processing center for the new refugees at Eglin Air Force Base in the Florida panhandle.

The influx of refugees continued yesterday as more small boats reached Key West, straining facilities there and overburdening the transportation system set up to transfer them to Elgin.

Officials estimated that more than 2,100 refugees reached Key West yesterday, bringing to more than 15,100 the total number of arrivals during the two-week sealift from Cuba.

Gov. Bob Graham, in an urgent letter to Carter, appealed for help to "avert impending disaster" and criticized the government for being "sluggish" in assuming responsibility for handling the refugee flood.

"There now exists a shortage of food, water and space for the refugees in Key West," Graham wrote Carter. "I ask you to declare an emergency exists in Florida."

"We have about 4,000 people here and we can only handle 2,000," an aide to Graham, Ron Villella, said earlier. "We're falling behind and we do not have the capability of feeding and housing this many people."

Federal and state officials were using fleets of buses and airplanes to transfer the new arrivals to the Eglin processing center, but said the transportation system was not adequate to handle the weekend flood.

At Eglin, the refugees were housed in a makeshift four-acre tent city a few miles from downtown Fort Walton Beach. Officials at the processing center said the resettlement of the refugees in other U.S. cities could begin as early as today.

In his speech to the League of Women Voters, the president appealed to Americans to remember their heritage in dealing with the Cuban refugees.

"We are the most generous nation on earth in receiving refugees and I feel very deeply that this commitment should be maintained," Carter said. "Our is a country of refugees. Many of those in this room have either parents or grandparents who were refugees, who came here looking for a new life of freedom, a chance to worship as they pleased, or a chance to combine their own talents to build a growing and dynamic country.

"Those of us who have been here for a generation or six or eight generations ought to have just as open a heart to receive the new refugees as our ancestors were received in the past."

Powell said the president's remarks did not signal a change in overall refugee policy, but an attempt "to deal with the situation as it exists."

In the coming weeks, he said, the administration hopes "to impress on all nations the fact that the plight of refugees is not a responsibility that should be borne by any one country."