When Laxmi Berwa of Clinton heard that India's Prime Minister Indira Gandhi was coming to Washington, he saw an opportunity to deliver a personal plea on behalf of India's most downtrodden class, its 14 million "untouchables." Berwa, a physician who is an untouchable, wrote Gandhi a letter asking to meet with her.

In a short reply, Gandhi chided Berwa for criticizing her government, adding that "it does not help for those who are living in affluence abroad to comment on situations about which they have little knowledge." Berwa was told Gandhi was too busy to see him.

Yesterday morning, Berwa and about 20 other Indian immigrants gathered outside the White House while Gandhi met inside with President Reagan. Like Berwa, the protesters were members of the untouchable caste, the lowest rung on India's social ladder. They were there to register their disapproval of what they see as the Gandhi government's indifference toward the caste.

Their protest will probably be regarded as a minor wrinkle in an otherwise smooth and cordial visit by the Indian prime minister to this country. Tomorrow she is scheduled to meet at the Kennedy Center with 2,500 members of the 22,000 Indians living in the Washington area in what is expected to be a very friendly affair. By contrast, Berwa estimates there are about 200 untouchables living in this area and about 4,000 in the United States and Canada.

But yesterday's demonstration touches a tender nerve in the Gandhi government because it relates to one of the most intractable social problems facing the prime minister, and one that causes embarrassment overseas.

For centuries, the untouchables, who make up one-seventh of India's population, have been regarded as "unclean" by their fellow Indians, assigned to the dirtiest, lowest paid jobs, shunned socially by other castes or classes of society, and neglected in social services and education. Their position at the bottom rung of Indian society was endorsed by the religious precepts of Hinduism, the dominant religion of India.

When India became independent in 1947 the government outlawed discrimination against the untouchables and embarked on an affirmative-action program by reserving a certain number of elected posts, civil service jobs and educational scholarships for them. The results, however, have been mixed.

Spokesmen for Gandhi say the government is doing the best it can to eliminate discrimination whose roots go centuries back in Indian society. "We don't deny the problem exists, but its existence spurs us on to efforts to eradicate this problem," said Indian Embassy press spokesman Deepak Vohra. "Some people say we do not do enough but it's a question of attitudes which are difficult to change overnight."

As an example of those efforts Vohra noted that India's ambassador to the United States, Kocheril Narayanan, is from the untouchable caste.

But progress in uplifting the untouchables has been spotty and critics like Virendra K. Chaudry, a New York engineer and of the untouchable caste, who was also at the demonstration yesterday said it is not enough. "It's eyewash," he said.

Some American scholars of Indian politics say that despite the laudable intentions of India's antidiscrimination laws, the implementation of those laws has been weak. They criticize a lack of governmental pressure to end an increasing number of violent attacks on untouchables in recent years, particularly in the countryside where 85 percent of them still live.

"We want to tell Mrs. Gandhi we are not living in the 18th century, we are in the satellite age, we know what is going on at home," said Berwa. "We really want to voice our concern about the atrocities on untouchables in India. It is our moral obligation to do it."