As in the case of Goldstone bosons, it is massless, unless there is, in addition, a small explicit supersymmetry breakdown involved, on top of the basic spontaneous breakdown; in this case it develops a small mass, analogous to that of Pseudo-Goldstone bosons of chiral symmetry breaking.
In theories where supersymmetry is a local symmetry, the goldstino is absorbed by the gravitino, the gauge field it couples to, becoming its longitudinal component, and giving it nonvanishing mass.
This mechanism is a close analog of the way the Higgs field gives nonzero mass to the W and Z bosons.
Vestigial bosonic superpartners of the goldstinos, called sgoldstinos, might also appear, but need not, as supermultiplets have been reduced to arrays.
[1][2][3] In effect, SSB of supersymmetry, by definition, implies a nonlinear realization of the supersymmetry in the Nambu−Goldstone mode, in which the goldstino couples identically to all particles in these arrays, and is thus the superpartner of all of them, equally.