How to get the effect of one dummy variable against many others?
If you only want to test whether the mean wages of South Dakotans is higher than for residents of other states, run:



Where South Dakota is a dummy variable like the one you've described earlier. I'm assuming you simply have a cross-section of wages at a single point in time, but if you have either repeat observations of individuals or a repeated cross-section, then you can include a t subscript for each of the variables and include time controls.  

If  is positive and significant, it would be interpretable as the level or percent (if the model is run in log wages) increase in mean wages for a resident of South Dakota compared to the mean for the pool of residents of all other states. 

If instead you wanted to perform pair-wise comparisons between South Dakotans and residents of an individual state, you could include a dummy for that state and then run a command like lincom in Stata to perform hypothesis tests between those states' coefficients.   

If you wanted to compare South Dakota against every single other state, then make South Dakota the omitted category in your regression, and you could run the model you described of including dummies for all non-South Dakota states. The big issue with this approach is the massive multiple hypothesis testing issue you will encounter, which could be addressed through Bonferroni correction.