A robust and simple mid-infrared hollow-core anti-resonant fiber (ARF) based refractive index (RI) sensor with an intensity demodulation method is presented and analyzed for monitoring liquid analytes. The ARF allows liquid analytes to flow through its hollow area for detection. To obtain ideal sensing performance, an epsilon negative (ENG) material is introduced into the selected anti-resonant tube. With the high absorption of the ENG material, only one fundamental mode is available for detection and is sensitive to the RI variation of analytes. Moreover, the effects of structural parameters on the sensing performances are discussed and analyzed to further understand the mechanism and optimization. The final result shows that the ARF sensor can exhibit a high sensitivity of −372.58 dB RIU−1 at a fixed wavelength within a broad RI range from 1.33 to 1.45, which covers most liquid analytes. It is a promising candidate for chemical and environmental analysis. Additionally, it has the potential for deep research to feed diverse applications.