Kinetic Alfven waves (KAWs) can play an important role in the energization of plasma particles and the formation of filamentous structures, which commonly are encountered and frequently accompanied by field-aligned currents in various magneto-plasmas, such as laboratory, auroral, and coronal plasmas. Based on a low-frequency kinetic dispersion equation in frequency omega < omega(ci) (the ion cyclotron frequency), KAW instability driven by a field-aligned current, which is carried by the field-aligned drift of electrons at velocity V-D, is investigated in a low-beta plasma of beta < Q << 1, where beta is the kinetic-to-magnetic pressure ratio and Q (equivalent to m(e)/m(i) << 1) is the mass ratio of electrons to ions. An instability condition and the corresponding growth rate are obtained, which depends on the plasma beta parameter as well as the drift velocity V-D. The results show that the KAW instability occurs in the perpendicular wave number range of k(perpendicular to)(-) < k(perpendicular to) < k(perpendicular to)(+), in which the growth rate reaches its maximum at k(perpendicular to)(m) for fixed V-D and beta. As V-D increases, this growing wave number range widens and the growth rate increases, but the maximal-growing wave number k(perpendicular to)(m) decreases. On the other hand, as the plasma beta parameter decreases, the growing wave number range also widens, and the maximal-growing wave number and growth rate both increase. These results have potential importance in understanding the physics of the electric current dissipation and plasma active phenomena since the field-aligned current is one of the most active factors in space and astrophysical plasmas.