We search for the $Z_1(4050)^+$ and $Z_2(4250)^+$ states, reported by the Belle Collaboration, decaying to $\chi_{c1} \pi^+$ in the decays $\bar B^0 \to \chi_{c1} K^- \pi^+$ and $B^+ \to \chi_{c1} K^0_S \pi^+$ where $\chi_{c1} \to \jpsi \gamma$. The data were collected with the BaBar detector at the SLAC PEP-II asymmetric-energy $e^+e^-$ collider operating at center-of-mass energy 10.58 GeV, and correspond to an integrated luminosity of 429 fb$^{-1}$. No evidence is found for the $Z_1(4050)^+$ and $Z_2(4250)^+$ resonances, and 90% confidence level upper limits on the branching fractions are reported for the corresponding $B$-meson decay modes. Show Partial Abstract