MONTEVIDEO, Uruguay— From the crack of bare hands driving the ball across the fronton court to the waitresses in green and red serving garlic-laden dishes, a modest downtown club is trying to keep alive an identity brought from the distant Pyrenees.

''I've been part of this since I was in short pants,'' said 66-year-old Jorge Arin Ayphassorho, the historian of Euskal Erria, or the Basque Institution. ''Being Basque is a credential of work and sacrifice here in Uruguay. We are trying to transmit those traditions.''

Although this immigrant nation is generally viewed as half Spanish and half Italian in origin, substantial numbers of those from Spain have ancestral roots in the autonomy-minded Basque provinces. Mr. Arin estimates that a fourth of Uruguay's three million people have at least one parent with a Basque surname.

Argentina and Chile, the two other countries that with Uruguay make up the so-called Southern Cone of South America, reflect a similarly strong Basque presence. While there is little scientific evidence to prove it, the Basque tie often gets the credit, or blame, for the strengths, weaknesses and peculiarities of the region. A Handshake Is Sufficient

''The Uruguayan always has something in him that is inherited from the Basques,'' Mr. Arin said. ''Here, for example, we talk about 'Basque's word.' That means to close a deal on a handshake, that it is not necessary to put it on paper.''

Patricio Aylwin, the front-runner in the Chilean presidential elections in December, who has a Welsh last name but counts more Basque ancestry, likes to recount the comment of a friend that ''the two greatest accomplishments of the Basques are the Republic of Chile and the Jesuits.''

The religious order is known in Central and South America for having a large number of Spanish Basques among its priests and for educating the upper classes and future leaders, like Luis Alberto Lacalle, the front-runner in Uruguay's presidential elections on Nov. 26. Mr. Lacalle is not a Basque, however.

Among the perceptions about the Basques in the Southern Cone -many, no doubt, self-created - are that the Basques are more honest, harder working, tougher and richer than the average. In Uruguay and Chile, for example, the notion exists that the Basques constitute a sort of ruling elite cutting across party and ideological lines and that that helps to explain why the two countries have far less official corruption and more capable bureaucracies than are the norm in Latin America.

That does not explain Argentina, which has lots of rich, old-family types with Basque surnames but is dogged by corruption and bureaucratic ineptitude.

It used to be that the Basque connection was taken as part of the explanation for the long traditions of democracy in Chile and Uruguay. Both countries suffered military coups in 1973. Uruguay returned to elected government in 1985 and Chile will have elections next month. Pinochet's Basque Background

That Gen. Augusto Pinochet Ugarte was one of the leaders of the Chilean military coup and has dominated the Government for 16 years is sometimes seen as evidence that Basques are tough and invincible, rather than a comment on their allegiance to democracy. He is French Basque on his father's side, Spanish Basque on his mother's.

Euskal Erria was founded in 1912 for descendants of Basques from both the Spanish and French sides of the region. It is an offshoot of another Basque organization in Montevideo, Centro Euskaro, founded a year earlier. Differences over the Carlist wars and the Spanish Civil War have kept the two clubs apart.

In addition to the fronton court, where handball, played either with the hand or a paddle, is popular with many non-Basques as well, the club gives lessons in the Basque language and organizes social activities. ''There is no Basque born in Uruguay who has not played ball with the bare hand,'' asserted Mr. Arin's brother, Roberto, president of the club.

Basques, he said, like to create sports that are ''close to work habits,'' such things as tree-cutting competitions or measuring a man's strength against oxen.

In 1923 the institution founded a secondary school for Basque descendants and brought Roman Catholic nuns from the Basque lands to run it. But it was closed five years later because the Uruguayans, who have always had rather liberal ideas about religion, decided the European nuns were, in Mr. Arin's words, ''very narrow'' in perspective.

The Basque provinces, nestled in the northeastern corner of Spain beside the Bay of Biscay and across the Pyrenees in France, sent small numbers of settlers to the New World in the earliest colonial days, but the first substantial Basque immigration came from the French side beginning about 1824. Mr. Arin said hunger and the shortage of farmland in the region brought those people to South America.

In the Spanish Civil War of 1936-39, the Basques supported the losing side and left Europe as a result. In addition to coming to the Southern Cone, many also went to Venezuela, Mr. Arin said.

While Euskal Erria claims to support the people of the Basque lands in Europe, Mr. Arin said it wants nothing to do with E.T.A., the guerrilla organization that seeks the region's full independence from Spain.

''Those are just young people with nothing more to do than go out and kill,'' he said. ''Basques don't accept that. They believe you are supposed to fight only if attacked first.''

photo of handball players in Montevideo (NYT/Eduardo Escudero)