The transmission reach of multi-core fiber (MCF) communication systems is severely affected by inter-core crosstalk (IC-XT), which limits its application for long-reach core optical network. One of the major factors limiting the transmission reach of MCF is nonlinear IC-XT interference, which makes the overall system nonlinear, thereby resulting in a poor bit error rate (BER) performance. Conventional Volterra series based nonlinear equalizer are computationally complex, and impaired by modeling error due to the truncation of polynomial kernel. In this paper, for the first time, we propose multivariate kernel least mean square (KLMS) based adaptive nonlinear equalizer for mitigating IC-XT impairments in MCF communication systems. The proposed scheme is inspired from reproducing kernel Hilbert space (RKHS) based machine learning algorithms. Simulations are performed for different multi-core structures, fiber lengths, and modulation schemes, which show that the proposed KLMS algorithm exhibit superior BER performance over the existing Volterra series equalizer.