A simple graph illustrating government's role in healthcare.

First of all, Libertarians generally advocate a return to Constitutional principles. Nowhere in the Constitution does it mention "healthcare" as a right of its citizens, nor that it be provided by the central government. Secondly, we actually have answers to its current failings. 

A common question posed by big-government advocates and Democrats specifically is, "What do YOU propose we do to fix healthcare?". It's normally met with silence on the other side of the aisle, mostly because of special interest donors to the Republican side. I can give you the Libertarian answer to this seemingly undefeatable question, keeping in mind the dramatic reduction of taxes in a libertarian paradigm:

• Establish Medical Savings Accounts: You could have the option to deposit tax-free money into a Medical Savings Account (MSA). When you need it, withdraw it. In addition, Libertarians will work to make all healthcare expenditures wholly tax deductible. Add that to the increase in take-home pay due to dramatically lower taxes, that Libertarian choice starts to look better and better.
• Deregulate the healthcare industry: All government policies that increase costs and decrease the availability of care must go. For example, mandating coverage of specific ailments, mandating minimum plans or prohibiting only catastrophic coverage, and "any willing provider" rules. States are the main culprit of these restrictions, so as usual, the choice is with the citizens of the offending states.
• Remove barriers to safe, affordable medicines: In short, cut the "red tape". Agencies like the FDA are antiquated, innovation inhibiting entities. The FDA drives up costs and delays life-saving treatments at a staggering rate. One example of this is was the 10 year delay to approve Propranolol (heart medication), in the mean time an estimated 100,000 people died that could have been treated by it. Libertarians would facilitate a transition to free-market alternatives for quality control studies and in-turn eliminate the burden of tax funding and watch-dogging; at the same time creating a whole new industry.
• Allow American citizens to purchase healthcare across State lines: Simply stated and it makes sense doesn't it? The proposed Bills in Congress don't address nor even sniff this as a possibility. Why? Even if it would be so simple to enact and would increase competition (read:reduce cost and increase service) overnight? Because it would actually mean the Federal government would be giving up a regulatory power they had successfully seized. Libertarian want virtually every regulation stifling free trade stripped from the books, thereby increasing the consumer's control over the cost and quality of the product, in this case healthcare. 

Below, linked, is a letter by a doctor in south Denver. Written in 2007, it's once again relevant; further proof of the fact that the bureaucrats won't stop their power grab.