I think the minimum wage should be increased, not necessarily to $15 an hour nationally, but it should definitely be adjusted at least to account for inflation since the wage was established.
Most arguments against minimum wage increases seem to stem from around 4 key arguments.
While yeas I believe raising labor costs will lead to automization the loss of jobs will not be 100%. While the fry cook may lose their job, now the company must pay someone to design/maintain/program/install etc. the machines that replaced the fry cook. Ditto for other fast food positions. Throughout history certain jobs have been eliminated only to be replaced by different positions and industries.
First off congratulations for doing that and I would pose the question that instead of paying fast food workers less maybe you should be paid more? If the wage floor increases it will force total wages to rise. There are still skill gates for what are currently higher paid positions. For example a EMT (throwing out a position that I think is underpaid) has a job that requires more skills than FF work. A cashier can't just say I want to be an EMT without passing through skill gates. As a result your value remains. Except now you can jump ship to say fast food work etc. and your pool of reasonable work has increased. Thus to maintain the employees they need the pay for EMT's must be increased as well, otherwise fewer people will pursue that career.
http://www.governing.com/gov-institute/voices/col-raising-minimum-wage-boost-middle-class.html
The problem is prices have already risen faster than real wages, working a minimum wage job today in most of the country is worse than it was 30 years ago. Prices might inflate, but I doubt they will exponentially increase. Plus most people in the lower income strata spend rather than save, which would likely impact many businesses that would be paying wages. Most minimum wage workers I know would probably be spending their wage increases at the places they work or similar.
This I feel is a more ideological disagreement. I feel that markets are imperfect and require government moderation. However I will bring up the point that the government already meddles in the market by providing social programs such as Medicaid/Food Stamps etc. This has ultimately allowed companies to subsidize minimum wage employees to an extent with taxpayer funds. If we raised minimum wage so the bottom tier of working poor no longer required the same extent of government assistance it would be a net gain. We would be turning people who are net drains on the economy (tax wise) and turning them into net contributors especially in regards to sales tax (because once again the poor spend most of what they earn rather than save due to necessity) and more importantly payroll taxes etc. will help fund programs facing a budget shortfall like Social Security.
Overall raising minimum wage I think would benefit the economy as a whole! Not to say people opposing it are evil I think for the most part there has been alot of anger. I have worked minimum wage jobs or close to it and they were hard... far harder than the accounting I am currently doing 49-$10 an hour to work (I am from Washington State) in a warehouse is exhausting more so than sitting in an air conditioned office crunching numbers... actually I crunched numbers while working inventory in a warehouse for less money and more discomfort.