We report the design and characteristics of a simple and compact mode-locked Er-doped fiber laser and its application to broadband cavity-enhanced spectroscopy. The graphene mode-locked polarization maintaining oscillator consumes less than 5 W of power. It is thermally stabilized, enclosed in a 3D printed box, and equipped with three actuators that control the repetition rate: fast and slow fiber stretchers, and metal-coated fiber section. This allows wide tuning of the repetition rate and its stabilization to an external reference source. The applicability of the laser to molecular spectroscopy is demonstrated by detecting CO2 in air using continuous-filtering Vernier spectroscopy with absorption sensitivity of 5.5 × 10−8 cm−1 in 50 ms.