A fixed wage floor subject to inflation is a temporary remedy that gradually evaporates. For opponents of a minimum wage, the absence of automatic adjustment in inflationary times is a de facto slow-motion repeal. Opponents would most prefer no minimum wage at all, but failing that would much prefer a fixed evaporating minimum wage to a permanent one.
As for the rest, the advantages and disadvantages of an automatically adjusted minimum resolve around two things:
Inflation itself. Employers who buy and sell things cope with inflation constantly, and adjust their purchasing choices and prices accordingly. Labor is in this sense just another purchased widget needed to do business, and shouldn't be that different from various other widgets.
Feasibility of initial minimum wage. If a 2018 automatically adjusted minimum wage was initially set to $1,000,000/hour, that either would ruin all, (or most), employers, or create a huge black market. If a 2018 automatically adjusted minimum wage was initially set to $5/hour it would be much the same as no minimum at all.
Somewhere in between there's a sweet spot, which might be a matter of experimental trial-and-error, and perhaps might also require adding some additional variable to the adjustment formula.
For experimental testing purposes, (i.e. properly measuring the results of a longstanding minimum wage), an automatic adjustment is preferable, as it would eliminate the variable of deflation. For example, supposing a too low deflated minimum correlated with working family malnutrition, an opponent might wrongly infer that it was the existence of any minimum at all which led to malnutrition, rather than a ruinously low wage. Similarly, an unfeasibly high wage floor might eventually deflate to an actual sweet spot with some positive results, which an advocate might wrongly infer proved the unfeasible wage was in fact the optimum minimum wage.