I'm working on a political campaign for a Democrat and I've heard people say "I don't vote Democrat, I work for a living," multiple times. There's also the election day joke "Sure Democrats are polling well now, but wait until people get off work" which is apparently very old. And apparently unemployment is rather evenly spread out between states: http://data.bls.gov/map/MapToolServlet?survey=la
So where did this idea that Democrats don't work come from? When did it start? Why has it persisted for so long?
Since the era of the New Deal, the Democratic Party has generally been identified with the notion of expanding the social welfare state. One of the most controversial measures -- generally supported by Democrats by generally opposed to different extents by Republicans -- has been the program today formally known as today as TANF (Temporary Assistance to Needy Families), which was formerly known as AIDC (Aid to Families with Dependent Children) before reforms were implemented in the mid-90s -- but which most people simply refer to by its more familiar name "welfare."
Essentially, the program gave monetary assistance to people with little-to-no incomes. At first, it had a very limited number of recipients, but as time grew the program morphed and the number of people receiving benefits from the program got larger and larger. Many people believed that welfare recipients were in large part lazy, and that the program's style of providing people who had no incomes with cash assistance discouraged these people from seeking or accepting jobs (why look for work when you get a check for free?). Many people who worked for a living felt (and still feel) outraged that their tax dollars, as they saw it, were going to subsidize the lifestyles of people who were perpetually on the program and who were not making any effort to seek work and become independent.
As I mentioned, the program underwent some dramatic changes in the 1990s, but the Democratic Party has generally been identified with supporting the style of benefits that existed before these reforms, which many Democrats in Congress opposed at the time. I believe that's where most of the "trope" comes from.