  Recent hydrodynamical simulations have shown that differentially rotating
neutron stars formed in core-collapse supernovae may develop global
non-axisymmetric instabilities even when $T/|W|$ (the ratio of the rotational
kinetic energy $T$ to the gravitational potential energy $|W|$) is relatively
small (less than 0.1). Such low-$T/|W|$ instability can give rise to efficient
gravitational wave emission from the proto-neutron star. We investigate how
this instability is affected by magnetic fields using a cylindrical stellar
model. Wave absorption at the corotation resonance plays an important role in
facilitating the hydrodynamic low-$T/|W|$ instability. In the presence of a
toroidal magnetic field, the corotation resonance is split into two magnetic
resonances where wave absorptions take place. We show that the toroidal
magnetic field suppresses the low-$T/|W|$ instability when the total magnetic
energy $W_{\rm B}$ is of order $0.2\,T$ or larger, corresponding to toroidal
fields of a few $\times 10^{16}$ G or stronger. Although poloidal magnetic
fields do not influence the instability directly, they can affect the
instability by generating toroidal fields through linear winding of the initial
poloidal field and magneto-rotational instability. We show that an initial
poloidal field with strength as small as $10^{14}$ G may suppress the
low-$T/|W|$ instability.
