The short answer is we couldn’t know as he never experienced or conceived of such a thing back in the 1870s. However he was a noted supporter of the ideas of the Rev. Thomas Malthus who had published his Essay on Population back in the late 18th century. Malthus believed that there was a natural tendency for populations (especially human) to outstrip food supplies and thereby the collapse of social order. It was therefore not desirable for the ruling classes to seek to prevent poverty, disease or war as these were natural restraints on the tendency for population to outstrip resources. Malthus, it seems, was personally a kindly, compassionate man but he was the architect of a philosophy that made expedient the hardship of a minority for the benefit of the majority living in a stable society. Politically Malthus’s ideas were seized-on to justify the ‘workhouse system of poor law’ instituted by the Poor Law Amendment Act of 1834 and which replaced the variable ‘Speenhamland system’ with a national system of regulation.
Darwin (and Malthus) were also concerned about the tendency for ‘inadequate’ members of society to reproduce beyond their capacity to support their dependents and to thus become a ‘brake’ on social progress. This line of thought became a justification for later notions of eugenics (the science of ‘improving’ a population as a means of improving society) which gained a notorious reputation up to the 1960s.
So an informed guess as to an answer to the original question would be that Darwin would not, despite himself being a caring father of 10 children and having a kindly personality, would not have supported the development of a modern welfare state.
Above illustration of Hull Royal Infirmary opened in the mid 1960s. It replaced the southern complex of one of Hull’s two Victorian workhouses. Part of the workhouse complex however was retained behind the modern block and was used as general wards up to a few years ago and latterly used as administration space. Sadly this remain part of the old workhouse has been demolished in the last few weeks and even more sadly I didn’t think to get a picture of it.