EDIT: New answer as of 10 Dec 2018.
Because the nominal variable has only two levels, you could use either Kendall or Spearman correlation. Readers would most likely be more familiar with these measures (tau or rho, respectively) than with alternatives.
Again, because there are only levels in the nominal variable, you could also use effect size statistics that might be used alongside a Wilcoxon-Mann-Whitney test. I find Vargha and Delaney's A easy to interpret, and some prefer the related Cliff's delta. There is also an r which is defined as the Z value from a WMW test divided by the total sample size. There is some information on these statistics on this webpage, of which I am the author.
In general, the degree of association between a nominal variable and an ordinal variable can be assessed with Freeman's theta or a statistic sometimes called epsilon-squared. Epsilon-squared is described here, and is pretty commonly spotted around the internet. Freeman's theta is mentioned around the internet, but I think for the calculations, you have to get the Freeman 1965 book or see the R function freemanTheta, of which I am the author. There is some information on these statistics on this webpage, of which I am the author.