Sensitive detection of terahertz fingerprint absorption spectrum for tiny molecules is essential for bioanalysis. However, it is extremely challenging for traditional terahertz spectroscopy measurement because of the weak spectral response caused by the large mismatch between terahertz wavelengths and biomolecular dimensions. Here, we proposed a wideband-tunable metal plasmonic terahertz biosensor to detect tiny biomolecules, employing attenuated total reflection in an Otto prism configuration and tightly confined spoof surface plasmons on the grooved metal surface. Benefitting from the plasmonic electric field enhancement, such a biosensor is able to identify the molecular terahertz fingerprints. As a proof of concept, a hypothetical molecule modeled by the Lorentz model with two vibrational modes is used as the sensing analytes. Simulation results show that the absorption of two vibrational modes of analytes can be selectively enhanced up to ten times by plasmonic resonance, and their fingerprints can be resolved by sweeping incident angle in a wide waveband. Our work provides an effective approach for the highly sensitive identification of molecular fingerprints in fields of biochemical sensing for tiny analytes.