  The two-channel photoproductions of $\gamma p \to K^{*+} \Sigma^{0}$ and
$\gamma p \to K^{*0} \Sigma^{+}$ are investigated based on an effective
Lagrangian approach at the tree-level Born approximation. In addition to the
$t$-channel $K$, $\kappa$, $K^*$ exchanges, the $s$-channel nucleon ($N$) and
$\Delta$ exchanges, the $u$-channel $\Lambda$, $\Sigma$, $\Sigma^*$ exchanges,
and the generalized contact term, we try to take into account the minimum
number of baryon resonances in constructing the reaction amplitudes to describe
the experimental data. It is found that by including the $\Delta(1905)5/2^+$
resonance with its mass, width, and helicity amplitudes taken from the Review
of Particle Physics [Particle Data Group, C. Patrignani {\it et al.}, Chin.
Phys. C {\bf 40}, 100001 (2016)], the calculated differential and total cross
sections for these two reactions are in good agreement with the experimental
data. An analysis of the reaction mechanisms shows that the cross sections of
$\gamma p \to K^{*+}\Sigma^{0}$ are dominated by the $s$-channel
$\Delta(1905)5/2^+$ exchange at low energies and $t$-channel $K^*$ exchange at
high energies, with the $s$-channel $\Delta$ exchange providing significant
contributions in the near-threshold region. For $\gamma p \to
K^{*0}\Sigma^{+}$, the angular dependences are dominated by the $t$-channel $K$
exchange at forward angles and the $u$-channel $\Sigma^*$ exchange at backward
angles, with the $s$-channel $\Delta$ and $\Delta(1905)5/2^+$ exchanges making
considerable contributions at low energies. Predictions are given for the beam,
target, and recoil asymmetries for both reactions.
