An integral domain $R$ for which the intersection of the nonzero prime ideals is nonzero is a [Goldman domain][1].  Equivalently: the fraction field $K$ is finitely generated as an $R$-algebra (equivalently, $K = R[f]$ for some $f \in K$).  The latter property is usually taken as the definition, but the equivalence is almost immediate: see e.g. $\S 12.1$ of [these notes][2].  Note also that the prominence of Goldman domains in commutative algebra is due as much to Kaplansky as to Oscar Goldman: under the name "G-domain", they play a surprisingly central role in his (perhaps slightly eccentric but very) influential text *Commutative Rings*.

For a general ring I don't quite know the answer to your question, but in his 1966 paper *The pseudo-radical of a commutative ring*, Robert Gilmer defines in any commutative ring the pseudo-radical to be the intersection of all nonzero prime ideals.  You can try to chase this down in the literature and see what you come up with. 

[![enter image description here][3]][3]


  [1]: http://en.wikipedia.org/wiki/Goldman_domain
  [2]: http://alpha.math.uga.edu/~pete/integral2015.pdf
  [3]: https://i.sstatic.net/BS6WQ.jpg