You can see Gallup's historical polling results on this question here: https://news.gallup.com/poll/4708/healthcare-system.aspx
Since 2000, the percentage of respondents who have said that providing healthcare coverage "is not the responsibility of the federal government" has fluctuated from a low in 2006 of 28% to a high in 2013 of 56%. (In their most recent survey, it's 42%.)