Mounting evidence suggests that luminous fast blue optical transients (LFBOTs) are powered by a compact object, launching an asymmetric and fast outflow responsible for the radiation observed in the ultraviolet, optical, infrared, radio, and X-ray bands. Proposed scenarios aiming to explain the electromagnetic emission include an inflated cocoon, surrounding a jet choked in the extended stellar envelope. Alternatively, the observed radiation may arise from the disk formed by the delayed merger of a black hole with a Wolf–Rayet star. We explore the neutrino production in these scenarios, i.e., internal shocks in a choked jet and interaction between the outflow and the circumstellar medium (CSM). If observed on axis, the choked jet provides the dominant contribution to the neutrino fluence. Intriguingly, the IceCube upper limit on the neutrino emission inferred from the closest LFBOT, AT2018cow, excludes a region of the parameter space otherwise allowed by electromagnetic observations. After correcting for the Eddington bias on the observation of cosmic neutrinos, we conclude that the emission from an on-axis choked jet and CSM interaction is compatible with the detection of two track-like neutrino events observed by the IceCube Neutrino Observatory in coincidence with AT2018cow, and otherwise considered to be of atmospheric origin. While the neutrino emission from LFBOTs does not constitute the bulk of the diffuse background of neutrinos observed by IceCube, the detection prospects of nearby LFBOTs with IceCube and the upcoming IceCube-Gen2 are encouraging. Follow-up neutrino searches will be crucial for unraveling the mechanism powering this emergent transient class.