An integral domain $R$ for which the intersection of the nonzero prime ideals is nonzero is a Goldman domain. 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. 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.
