By and large I believe most people's answer would be yes, but don't you dare raise MY taxes to do it. That seems to be the common theme. Look at people attitudes "what is the government going to do to help me?" There are many points to both sides of the issue, but is a tax shoulder by smokers really fair? Wouldn't a very, very small federal sales tax on all goods generate more money?
Good question.
It's hard to reconcile principles in this case. I personally think that as a relatively large government mandate it should come out of the general fund, and paid for out of personal income and corporate income tax. On the other hand, I'm neither deaf nor blind, and I know that a lot of general fund expenditures are simply unfunded and paid for out of borrowed money, which I'm not 100% keen on, and you shouldn't be, either, since the US federal debt has ballooned by 80% over the last 8 years.
Expanding the tobacco excise tax was a) an attempt to apply fiscally conservative pay as you go principles and b) fairly shrewd political gamesmanship. Now that one party holds congress and the white house, they don't need to play games and target tobacco explicitly. They might. But it's by no means guaranteed.
If given the option (ha! unlikely story.) I'd favor a segregated surtax based on income, which is progressive, then a sales, excise, or other consumption tax, which is in theory proportional (like a "flat tax") but in practice regressive, then a tax on any particular category of products, which is overwhelmingly regressive.
You can also look at this debate from another, more philosophical perspective, which I think that you were leaning towards. Yes, everyone is in favor of increasing services if they don't have to pay for them. That's because we have a transactional relationship - fee for service - with government. Government's no longer seen as an effective actor; it can't do anything right; all it does is fritter money away. In essence, there's no trust to do things right, and therefore no credit in situations where it can do something reasonably effectively, like SCHIP. So there you are, totally screwed again.