Increasing the minimum wage means increasing production costs. When production costs increase, the price of goods in the market increases, and therefore inflation increases. Since production costs increase, imported goods start to be purchased. As a result, your domestic production suffers and you become dependent on foreign countries because you are purchasing imported goods.
Basically, it will help the people to cope up with the increase on the price of marketable goods. Analogy is that;
If you're an average employee perhaps you are earning $15 on third world countries (at my area at least). Your usual daily expenses at lower inflation rate is around $12.
When inflation became more evident your daily expenses became $15, that leaves you nothing to save. Increasing the minimum wage will logically give people a margin. However it won't be sustainable to the economy of a country if GDP won't increase as well. To cut it short, it will be a whole lot of process and the thing we're discussing won't stop the struggle but will just temporarily aid it under conditions.