I think it depends. If a state subsidizes tuition to a public college, then the subsidy will keep the tuition rate down. What it will do, though, is shift resources from others areas of the state's budget to the university or college. Personally, I wish my state of Mississippi would spend more money funding technical training at the community college level in the aerospace and automotive industries instead of on things like the law school at Ole Miss. If someone wants to become a lawyer let him pay for it. It's not like these lawyers give citizens a break on fees because they got a bargain-priced education. What we need are fewer lawsuits and more jobs to complement the ones provided by companies like Northrop Grumman, Rolls-Royce, Boeing, Nissan, Toyota, American Eurocopter, Lockheed Martin, Raytheon, and Pratt & Whitney. In order to do that, we need a well-qualified technical labor force, and community colleges are, I think, up to the task--if they can get the money.
When it comes to private colleges, I don't think subsidies keep tuition down at all. Tuition is one of those things that's outpaced the rate of inflation, and it seems as though the more the government provides loans, grants, and tax credits to kids to attend college the more schools raise it. (I realize many schools provide their own primarily need-based financial aid as well.) At top-tier schools, people will pay whatever the market will bear, and the schools know it. The question is whether it's an efficient use of public resources in the form of grants, loans, and tax credits so people can attend Harvard, MIT, and Yale when they have applicants up the wazoo already. I don't think it is.