  Using $2.93~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data collected at a
center-of-mass energy $\sqrt{s}=3.773$ GeV with the BESIII detector operating
at the BEPCII collider, we search for the semileptonic $D^{0(+)}$ decays into a
$b_1(1235)^{-(0)}$ axial-vector meson for the first time. No significant signal
is observed for either charge combination. The upper limits on the product
branching fractions are ${\mathcal B}_{D^0\to b_1(1235)^- e^+\nu_e}\cdot
{\mathcal B}_{b_1(1235)^-\to \omega\pi^-}<1.12\times 10^{-4}$ and ${\mathcal
B}_{D^+\to b_1(1235)^0 e^+\nu_e}\cdot {\mathcal B}_{b_1(1235)^0\to
\omega\pi^0}<1.75\times 10^{-4}$ at the 90\% confidence level.
