We report on multiwavelength target-of-opportunity observations of the blazar PKS 0735+178, located 2.2 degrees away from the best-fit position of the IceCube neutrino event 211208A. The source was in a high-flux state in the optical, ultraviolet, X-ray, and GeV gamma-ray bands around the time of the neutrino event, exhibiting daily variability in the soft X-ray flux. The X-ray data from Swift-XRT and NuSTAR characterize the transition between the low-energy and high-energy components of the broadband spectral energy distribution, and the gamma-ray data from Fermi-LAT, VERITAS, and H.E.S.S. require a spectral cut-off near 100 GeV. Both measurements provide strong constraints on leptonic and hadronic models. We analytically explore a synchrotron self-Compton model, an external Compton model, and a lepto-hadronic model. Models that are entirely based on internal photon fields face serious difficulties in matching the observed spectral energy distribution (SED). The existence of an external photon field in the source would instead explain the observed gamma-ray spectral cut-off in both leptonic and lepto-hadronic models, and it would allow a proton jet power that marginally agrees with the Eddington limit in the lepto-hadronic model. A numerical lepto-hadronic model with external target photons reproduces the observed SED and is reasonably consistent with the neutrino event despite requiring a high jet power.