We systematically examine the fundamental frequencies of shear torsional oscillations in neutron star crusts in a manner that is dependent on the parameter L characterizing the poorly known density dependence of the symmetry energy. The identification of the lowest quasi-periodic oscillation (QPO) among the observed QPOs from giant flares in soft-gamma repeaters as the l = 2 fundamental torsional oscillations enables us to constrain the parameter L as L >= 47.4 MeV, which is the most conservative restriction on L obtained in the present work that assumes that the mass and radius of the flaring neutron stars range 1.4-1.8 M-circle dot and 10-14 km. Next, we identify one by one a set of the low-lying frequencies observed in giant flares as the fundamental torsional oscillations. The values of L that can reproduce all the observed frequencies in terms of the torsional oscillations coupled with a part of dripped neutrons via entrainment effects are then constrained as 101.1 < L < 131.0 MeV. Alternatively, if only the second lowest frequency observed in SGR 1806-20 has a different origin, one obtains relatively low L values ranging 58.0 < L < 85.3 MeV, which seem more consistent with other empirical constraints despite large uncertainties.