Pearson correlation method using absolute values and relative values
In general, the correlation coefficient is "invariant to separate changes in location and scale in the two variables". In particular, you can mix relative with absolute values. 

However, that only works if you scale the variables globally. You can't scale every individual data point (here on a city level). If this was a county wide election, you could scale the city values by the county population. 

But it sounds like your crime rates are on a per city level. In this case you should scale the votes on a city level as well to make them comparable. This will change the correlation coefficient and give a different result than with absolute values. I think using percentages is more intuitive in your case.