The internally contracted multi-reference configuration interaction method (MRCI) with Davidson modification and the Douglas-Kroll scalar relativistic correction is used to calculate $BSe^+$ ion at the level of aug-cc-PVQZ basis set. The calculated electronic states, including four triple and two quintuple $Λ$-$S$ states, are correlated to the dissociation limit of $B^+(^1S_g)+Se(^3P_g)$ and $B(^2P_u)+Se+(^4S_u).$ The states of $^3\Pi$(II) and $^3Σ^-$(II) from the dissociation limit of $Se^+(^4S_u)+B(^2P_u)$ both have double well and spectroscopic properties are studied. Various curve crossing are revealed, which could lead to the predissociation of the $X^3\Pi$ and $^5\Pi$ states and the possible predissociation pathway are analyzed. Spin-orbit coupling interaction is taken into account via the state interaction approach with the full Breit-Pauli Hamiltonian operator, which causes the entire six $Λ$-$S$ states to split into 21 Ω states. This is the first time spin-orbit coupling calculation of $BSe^+.$ The potential energy curves of the $Λ$-$S$ and Ω electronic states are depicted with the aid of the avoided crossing rule between electronic states of the same symmetry. Then the spectroscopic constants of bound $Λ$-$S$ and Ω states were obtained, which have never been observed in experiment.