Democrats may support some policies that lead to big government, however they don't support big government as inherently good, but merely as a unavoidable side effect to other important policies. Republicans, on the other hand, frequently argue that small government is worthwhile for it's own sake.
I think if we look at both motives without attempting to ascribe negative connotation, we might get somewhere.
"Small government" isn't purposeless, it's (supposed) tenet is to limit the government in both scope and authority to those powers enumerated by the Constitution. Either way, the point isn't just that the government should be non-existent, or smaller for its own sake, but that it should allow the citizens to provide for their own welfare and provide the minimal in opposition to that. The Republican mantra believes that people can and will fend for themselves, and either enjoy their own successes or failures with minimal intervention.
The democratic mantra is to provide a minimum standard for all, at the expense of those in excess of that standard.
It's a contrast in ideology, but neither is good or bad on its own. Also worth noting is that over time, the parties have effectively flipped positions on matters entirely. The Republicans, for example, freed the slaves despite FIERCE opposition by the democrats. Nowadays, the Democrats generally carry the African-American vote.
In short, it's a popularity contest. Both parties have ideals that they live up to some percentage of the time, but any individual politician (or party dynamic) is likely to shift their belief in the interests of getting elected, staying elected, appeasing campaign contributors or even to appease their constituency.
