's true that Democrats aren't picking these fights-- unpopular fights to pick in a country still as religious as the US-- but they're much more likely to support these things than Republicans. The California Democratic Party voted not to endorse the recent marijuana legalization bill, but only by a narrow margin and only because it could hurt their candidates in the election. Give it a few years.
It's not that the Democrats are carefully calculating not to pick that fight. It's that the vast majority of Democratic leadership favors the drug war. Marijuana legalization, the easiest part of that fight, is the tip of the iceberg. It's true that you'll find more Democrats than Republicans supporting ending the drug war, but not to such an extent that it makes sense to say they collectively favor "privatized morality." This is especially true when you realize that a substantial % of Democrats who favor at least scaling back the drug war don't want to make drug use a private decision, but rather want to force people into treatment programs instead of jails.
And, on a side note, there are two mainstream Republican candidates for the party nomination this cycle who favor ending the drug war. Neither will come close to winning, but that's two more than the Democrats had last cycle. The gap between the two parties might not be what you think it is, depending on what you think. That's because of the continued existence of more libertarianish Republicans on the edges of the party.
It's true that Democrats love to promote health and safety through taxes, product safety standards, and advertising campaigns. But public health is distinct from morality, and in any case, this sort of regulation is often done as an alternative to legislation.
I don't think you can come up with a justification for public health concerns that doesn't implicate morality. Anytime you make the statement, "The law out to be x" you are making a moral claim. I'm not sure what you mean by "alternative to legislation" when things like seat belt laws and smoking bans in private establishments are legislated.