The Spectator (U.K.) has an article by Dennis Sewell called "How eugenics poisoned the welfare state" that lists a number of prominent left of center folks who were ardent supporters of eugenics policies:
A century ago many leading leftists subscribed to the vile pseudo-science of eugenics, writes Dennis Sewell, and the influence of that thinking can still be seen today ...
Eugenics had been the brainchild of Charles Darwin’s cousin Francis Galton, and was developed in response to Darwin’s theory of natural selection. It was taken up as a programme of political action by Darwin’s son Leonard. The eugenicists aimed to replace natural selection with a planned and deliberate selection. They were alarmed by the fact that the poorest in society bred faster than the middle class, forecasting that this trend would lead to a spiral of degeneration in the gene pool. Their aim was to encourage the rich to have more children and the poor to have fewer. ...
For the Fabians, eugenics was not merely some eccentric hobby or sideline, but central to their social thinking. Beatrice Webb regarded eugenics as ‘the most important question’ of all, while her husband revealed the statist and dirigiste character of the movement with his declaration that ‘no eugenicist can be a laissez faire individualist… he must interfere, interfere, interfere!’ Even for George Bernard Shaw, ‘the only fundamental and possible Socialism’ was ‘the socialisation of the selective breeding of Man’. ...
Another Fabian eugenicist, the writer H.G. Wells, vented his frustration and indignation in a direct address to the working class. ‘We cannot go on giving you health, freedom, enlargement, limitless wealth, if all our gifts to you are to be swamped by an indiscriminate torrent of progeny,’ he complained, ‘...and we cannot make the social life and the world-peace we are determined to make, with the ill-bred, ill-trained swarms of inferior citizens that you inflict upon us.’ It was as if — as in the Brechtian joke — the Fabian left had lost confidence in the people and had determined to dissolve the people and appoint a new one. ...
The article skips some others, such as Winston Churchill during his Liberal Party years, Harold Laski (Britain's most prominent Jewish intellectual and chief ideologist of the Labour Party, he had been the golden boy protege of the elderly Galton), and statistician Karl Pearson (who changed the spelling of his first name from Carl to Karl to honor you-know-who). One interesting aspect of the article was a late enthusiast:
Eugenics was no quickly passing fad. The Eugenics Society reached its peak, in terms of membership, during the 1930s, and the cusp of the following decade saw the zenith of its prestige. The economist John Maynard Keynes served on the society’s governing council and was its director from 1937 to 1944. Once again, this was no casual hobby. As late as 1946 [the year of his death], Keynes was still describing eugenics as ‘the most important and significant branch of sociology’
Of course, now that I think about it, it's not at all surprising. After all, the distinguished Keynes and Darwin families intermarried, in the Galtonian manner: actor Skander Keynes, who plays Edmund in the Narnia movies, is a direct descendant of Darwin.
The funny thing is that Britain was just about the only advanced nation that didn't pass a law calling for the sterilization of mentally retarded people in the 20th Century. (The very progressive Swedes were doing this into the mid-1970s.) Why not? Largely, because another one of Darwin's relatives, a member of the Wedgwood family, took a strong stand against it in the House of Lords.
As for the point of Sewell's article in the Spectator, as best I can make out, he's arguing that the left of a century ago were worried that subsidizing the poor would create more of them, so they hoped that limitations on reproduction would heal the welfare state's Achilles heel.
It was during the late 1930s that much of the detailed planning for the welfare state was carried out. And a good deal of it was undertaken at meetings of the Eugenics Society. On the evening that the House of Commons met to debate the  Beveridge Report [outlining the post-WWII welfare state], Beveridge himself went off to address an audience of eugenicists at the Mansion House. He knew he was in for a rough ride. His scheme of family allowances had originally been devised within the Eugenics Society with a graduated rate, which paid out more to middle-class parents and very little to the poor. The whole point was to combat the eugenicists’ great bugbear — the differential birth rate between the classes. However, the government that day had announced a uniform rate. Beveridge was sympathetic to the complaints of his audience and hinted that a multi-rate system might well be introduced at a later date.
Which, presumably, never happened.
Sewell sums up his indictment of the welfare state. Unlike older conservatives who felt that the ideologues of the welfare state had been too optimistic about how well the work ethic would survive down through the generations under a welfare state, David Cameron's New Conservatives feel that the leftists of a century ago were too damn realistic:
Given the association of so many of its founding fathers with the dismal pseudo- science of eugenics, perhaps we should not be surprised that our welfare system has ended up preferring safety nets to trampolines [ouch], or that it prefers simply to warehouse the poor rather than give people who have fallen on hard times a chance to take responsibility for their own lives. Eugenics infected its adherents with a deeply pessimistic view of the poor, branding them as irredeemably genetically second-rate, and this view has cast a long shadow over social policy assumptions. Labour figures who mock the idea of ‘compassionate Conservatism’ or make light of David Cameron’s focus on our ‘broken society’ need to take a hard look at some of their own history and intellectual heritage. When it comes to who really can claim to care about the problems of the poor, the dividing lines are not so straight as Gordon Brown thinks they are.