By making wages higher you are helping to increase the incentive to work. Before the minimum wage came along, lots of people were not interested in getting off of benefits back and into employment simply because they would be going into badly paid jobs which didn't really have enough of an incentive to make them want to be coming off of benefits. This is called voluntary unemployemnt. Obviously by making a minimum wage which is more than benefits, you are making it worthwhile for people to actually go out and look for a job so they can have a higher income.
The knock on effect on the economy. There are two ways of looking at this, but one says that by increasing the level of disposable income at the lower end of the wage scale, you provide a significant increase to the economy. Basically, because people have more money, they spend more money. That in turn leads to increased revenues for small businesses, who then spend more money, which then increases the revenue for other businesses. This is what Economists call 'The Multiplier Effect'.
Against
Look at your a normal Supply/Demand curve which is used to determine the price for something. In the working market that price is the price of work, i.e. a wage. By increasing price you are raising the supply curve with the result that supply of labour expands, however, demand falls. In real terms this is easy to see. Increasing work costs often mean a lack in the amount of jobs available. This causes many people to believe that a minimum wage actually adds to poverty by pricing people out of jobs and increasing unemployment.
The knock-on effect on the economy. One half of this contributes to the 'For' argument but the other half contributes to the 'Against' argument. By increasing the wages of people at the bottom of the pay scale, this can cause businesses into having to increases the wages of the entire workforce in order to keep the difference between wages for different roles at the same or similar level. 'The Multiplier Effect' says that at basic levels this is a good thing, when taken further this can cause an increase in spending throughout the economy which can lead to rapid inflation (which has it's own knock on effects).
Conclusion
We don't however want the minimum wage to always stay the same. As the cost of living and wages in general rise, the minimum wage would also have to increase in order to give an incentive to stay in employment rather than living off benefits. A 'sensible' minimum wage can help to stabilize the economy by encouraging people back ito the workplace. If people can earn more by working than they would recieve on benefits, they wil be keener to find a job. Jobs are created when businesses produce goods and services that people want at prices they can afford. The more people in (reasonably payed) employment, the more good and services they can afford to buy, the more jobs created and so on. Also, the more people there are in employment, the less the government will have to pay in unemployment benefits. Another positive point about the minimum wage is that it encourages young people into the workplace. If young people can find a job paying a reasonable wage when they leave school and realise that they will be better off working than on benefits. If they start off by living on benefits they are more likely to stay on benefits and be unable to see the positive side of working. However, with a minimum wage the employers have to charge more for thier good or service to compensate for the raise in employee pay. They would then sell less of their good or service and be less competitive.