  M-dwarf stars are promising targets for identifying and characterizing
potentially habitable planets. K2-3 is a nearby (45 pc), early-type M dwarf
hosting three small transiting planets, the outermost of which orbits close to
the inner edge of the stellar (optimistic) habitable zone. The K2-3 system is
well suited for follow-up characterization studies aimed at determining
accurate masses and bulk densities of the three planets. Using a total of 329
radial velocity measurements collected over 2.5 years with the HARPS-N and
HARPS spectrographs and a proper treatment of the stellar activity signal, we
aim to improve measurements of the masses and bulk densities of the K2-3
planets. We use our results to investigate the physical structure of the
planets. We analyse radial velocity time series extracted with two independent
pipelines by using Gaussian process regression. We adopt a quasi-periodic
kernel to model the stellar magnetic activity jointly with the planetary
signals. We use Monte Carlo simulations to investigate the robustness of our
mass measurements of K2-3\,c and K2-3\,d, and to explore how additional
high-cadence radial velocity observations might improve them. Despite the
stellar activity component being the strongest signal present in the radial
velocity time series, we are able to derive masses for both planet b ($M_{\rm
b}=6.6\pm1.1$ $M_{\rm \oplus}$) and planet c ($M_{\rm c}=3.1^{+1.3}_{-1.2}$
$M_{\rm \oplus}$). The Doppler signal due to K2-3\,d remains undetected, likely
because of its low amplitude compared to the radial velocity signal induced by
the stellar activity. The closeness of the orbital period of K2-3\,d to the
stellar rotation period could also make the detection of the planetary signal
complicated. [...]
