The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given.