It's more complicated than this. Here is the chain of economics that makes this a complicated discussion:
1) We raise minimum wage
2) Because we raise minimum wage, more money enters the economy as more people spend. Inflation occurs over time.
3) As this inflation occurs, the value of the minimum wage increase decreases until (possibly) it is completely nullified. Poor people end up with the same total purchasing power, just making 15 dollars an hour to buy a 15 dollar T-shirt instead of making 8 dollars an hour to buy the same T-shirt for an uninflated 8 dollars.
Now, that possibly in italics in step 3 is the real discussion here. Virtually all economists agree that an increase in minimum wage causes an increase in inflation; the question is how large it is, and what portion of the increase in wages is nullified by it.
Given that America currently has an inflation rate below its target of 2-3 percent (I believe it's about