Britain's new Social Democratic Party will test its popularity with voters for the first time this summer when one of its founders, Roy Jenkins, a former Labor Party deputy and leader and cabinet minister, seeks election to a recently vacated seat in Parliament.

Jenkins, one of the new party's four coleaders, offered himself today as the Social Democratic candidate in a by-election in Warrington, an industrial city midway between Liverpool and Manchester in northwest England. Jenkins also will be supported by the Liberal Party in the first agreement of electoral cooperation between the old and new parties in the center of the British political spectrum.

The Warrington by-election, expected to be held next month, will pose an important and formidable challenge to the aspirations of the Social Democrats and Liberals to provide Britain with an electrable alternative to the right-wing policies of Conservative Prime Minister Margaret Thatcher and the leftwind drift of the opposition Labor Party.

With an aging working class population, Warrington has been a safe seat for the Labor Party, from which Jenkins and all but one of the 14 Social Democrats already in Parliament have recently defected. The Labor member of Parliament who resigned the Warrington seat last month to become a judge won 62 percent of the vote in the last election.

"I would propose to mount the most effective possible campaign demonstrating both the relevance of the Social Democratic Party in the problems of Warrington and the National reasons for the emergence of the party," Jenkins said, referring to the severe economic problems plaguing industrial Britain.

David Owen, a former Labor foreign minister who is now the leader of the Social Democratic group in the House of Commons, recently told a group of U.S. correspondents here that the new party's chief problem was winning bed-rock Labor seats, particularly in northern Britain.

"We will stand or fall on our success in taking seats from Labor in the north," Owen said.

Opinion polls and the new party's growing membership show that the Social Democrats are catching on most quickly in southern England and among middle class Britons, currently the stronghold of Thatcher's Conservatives.

"We could do quite well in the south and with the unions, the middle class and intellectuals and still not win many seats in Parliament," Owen said.

Political commentators here said it was important for the Social democrats that Jenkins finish at least a strong second in Warrington, beating out the Conservative candidate and winning a respectable share of the Labor vote.This also is an important personal challenge for Jenkins, who, like the new party he helped conceive, is accused by critics of being too hourgeois and intellectual to appeal to the working class voters the Social Democrats must woo away from Labor.

Jenkins, 60, had been a Labor member of Parliament for 28 years and served as home secretary and chancellor of the exchequer in past Labor governments. He played an important role in liberalizing British criminal laws and rallying political support for Britain's entry into the European Common Market.

With Owen and two other former Labor cabinet ministers, Shirley Williams and William Rodgers, Jenkins left labor and founded the Social Democrats.

Williams, the most popular member of the Social Democratic leadership, according to opinion polls, was first urged to run for the Washington seat but she refused. She had lost her seat in Parliament in the 1979 election.

The Social Democrats, according to one of their senior strategists, are pinning their hopes on having a number of by-election opportunities to build their strength and demonstrate their popularity before the next national election in 1984. Only two parliamentary seats have been by death or resignation so far since Thatcher's Conservatives won control of Parliament in 1979.

The popularity of a centrist Social Democratic-Liberal electoral alliance has slipped slightly in opinion polls since the heavily publicized formal launch of the Social Democratic party in April. The Warrington by-election will again focus attention on its rivalry with Labor, whose ascendant left-wing is working to make its policies, leadership and parliamentary candidates more militantly socialist.

But many Conservative politicians fear their party may be even more vulnerable than Labor to Social Democratic inroads because of the failure of Thatcher's policies to revive the ailing British economy and her unwillingness to modify them significantly. In wining the last election, the Conservatives hold many marginal seats in regions of the country hard hit by Britain's deep recession.

Without spelling out many specific policy proposals, the Social Democrats have said they favor maintaining Britain's mixed economy and welfare state social programs, breaking down class barriers and ending political polarization, and staying in NATO and the Common Market. They accuse Thatcher of eroding the welfare state and mismanaging the economy and Labor of wanting to nationalize most industry and pull out of NATO and the Common Market.