Summary: We obtain a formula for the number of rational degree \(d\) curves in \(\mathbb{P}^3\) having a cusp, whose image lies in a \(\mathbb{P}^2\) and that passes through \(r\) lines and \(s\) points (where \(r+2s =3d+1)\). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in \(\mathbb{P}^2\), which has been studied earlier by Z. Ran [Math. Proc. Camb. Philos. Soc. 127, No. 1, 7–12 (1999; Zbl 0972.14040)], R. Pandharipande [Trans. Am. Math. Soc. 351, No. 4, 1481–1505 (1999; Zbl 0911.14028)] and A. Zinger [J. Differ. Geom. 65, No. 3, 341–467 (2004; Zbl 1070.14053)]. We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger [loc. cit.] and I. Biswas et al. [J. Singul. 17, 91–107 (2018; Zbl 1423.14308)]. We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author [“Counting planar curves in \(\mathbb{P}^3\) with degenerate singularities”, Preprint, arXiv:2007.11933], where they compute the characteristic number of \(\delta\)-nodal planar curves in \(\mathbb{P}^3\) with one cusp (for \(\delta \leq 2)\).