The Democrats now in charge of Congress say that they want to make a higher minimum wage a "top priority" in the next session. The rhetoric usually centers on how difficult it is for anyone working full-time to raise a family on the equivalent of the minimum wage. Unfortunately, raising the minimum wage would be more politically popular than effective.
Problem #1: Raising the minimum wage is an artful dodge of solving real problems
Raising the minimum wage is a way for politicians to appear as though they're making poor people better off at the expense of the rich. It allows them to take credit for welfare that they force businesses to pay for -- thus creating a sort of welfare without taxation. But raising wages (by force of law) for 17-year-old suburbanite kids working at electronics stores in shopping malls is hardly the best way to address the real problems of poverty, like access to capital and educational opportunities.
Problem #2: Laws keeping minimum wages too high can keep kids from getting good jobs later
A higher minimum wage tends to keep kids out of work, since they're the ones least likely to have marketable job skills and thus have the least to offer in the labor market. The problem with that is it keeps those kids from gaining those entry-level experiences that make them more marketable.
If you're going to college, or if you have the right connections, that may be no big deal -- but if you're a 16-year-old from a low-income, low-education background, what you need more than anything else is an entry-level chance to get a start in the job force. It's much better for a 16-year-old to get a low-wage entry-level job while living at home with his or her parents than to have to wait to get job experience until after graduating from high school and moving out of the home (if, in fact, the individual graduates -- 15% of American adults haven't graduated from high school).
Problem #3: Government welfare mandates can lead to social unrest and crime
This is directly related to the first problem: Minimum-wage laws tend to create unemployment among the young. When legitimate employment opportunities for young people evaporate, they have more free time...but the same level of demand for goods, services, and amusement. In France, the government's overambition to supply an ideal level of welfare to all created a huge class of unemployed youth who turned to violence, partially over their anger about their lack of access to jobs.
While the United States has had fewer riots among unemployed youth lately than France, many young Americans with few job opportunities have turned to criminal behavior, like the sale of illegal drugs -- a dangerous and (surprisingly) low-income alternative. This isn't to say that every member of a drug gang would prefer to work in a $5 an hour entry-level job in the legitimate sector, but undoubtedly some would take the opportunity if it existed.
Problem #4: An effective minimum wage makes goods more expensive for low-income workers
Assuming (conservatively) that labor costs make up about half of the cost of doing business in the United States, then if the cost of labor is forced up for low-wage jobs, the customers of those firms using low-wage labor will end up paying more to get the same things.
But which industries are heavy users of low-wage labor? Among others, leisure and hospitality (including restaurants and hotels), retail, and textile/apparel production. Within those industries, it's safe to assume that those restaurants, stores, and clothing makers that cater to higher-income consumers already tend to pay relatively higher wages to their employees -- the wait staff at Ruth's Chris or the line workers making Gore-Tex aren't going to be affected directly by a rising minimum wage. But if wages (and costs) are forced upwards by law at convenience stores, deep-discount retailers, and fast-food restaurants -- the kinds of places making up a larger share of spending for lower-income workers than higher-income workers -- then the customers shopping there will have relatively less buying power. To get their "low prices," Wal-Mart customers depend upon (relatively) low wages for Wal-Mart employees.
Problem #5: An effective minimum wage can put people out of work
An effective minimum wage -- that is, one which causes some employers to have to pay more -- often creates unemployment. Since time spent out of work is oftentimes time spent not developing new marketable job skills, unemployment is actually a double-whammy for the unemployed individual. It not only is time spent not making an income, it's also lost opportunity to keep up with the skills that other workers are developing.
Problem #6: Minimum-wage laws aren't as effective as negative income taxes or the EITC
Raising the minimum wage is usually championed as a means of helping the poor. But as Milton Friedman championed, it's much more efficient to establish a negative income tax (or, in the US, the Earned Income Tax Credit). A higher minimum wage tends to keep teenagers out of work and prevents many people from making extra money by taking up low-wage second jobs. An Earned Income Tax Credit instead focuses on those who are actually low-income and avoids many of the secondary unfortunate consequences of the minimum wage.