I was just reading the NYT online when I came across this article referring to Michael Bloomberg as a "democrat" in Republican clothing. While I've given up being conned by any notion that either of the bourgeois political parties actually represents the interests of the working class, it does seem to me that this article provides us with a good way to talk about what the Democrats once were and what they've become. The issues that the author of the Bloomberg article describes that make Bloomie a "Democrat" are gun control, stem cell research, more generous public school funding than George Pataki, and endorsing Democrat in a state senate race. With the exception of school funding, these issues are mighty particular - and cultural ones - on which to define party lines. Even in the age of Jackson, the one of the primary constituencies of the Democratic party was white organized labor. Since the 1930s rise of the CIO and the New Deal, you can add Black urban workers to that constituency, and after 1964, you can add African Americans more generally.
Bloomberg's behavior towards municipal and other workers marks him as an arch-conservative plutocrat, and in my opinion, that's more important than Stem Cells as a dividing line. If being a democrat only means opposing the evangelical right wing (stem cells), then being a Democrat today means being slightly to the right of those termed "Rockefeller Republicans."