A class of thick branes in the background of sine-Gordon kinks with a scalar potential V(φ)=p(1+cos) was constructed by R. Koley and S. Kar [Classical Quantum Gravity 22, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin-1=2 fermions on these branes in the presence of two types of scalar-fermion couplings: and sin. By presenting the mass-independent potentials in the corresponding Schro¨dinger equations, we obtain the lowest Kaluza-Klein modes and a continuous gapless spectrum of Kaluza-Klein states with m2>0 for both types of couplings. For the Yukawa coupling, the effective potential of the right chiral fermions for positive q and is always positive; hence only the effective potential of the left chiral fermions could trap the corresponding zero mode. This is a well-known conclusion which is discussed extensively in the literature. However, for the coupling sin, the effective potential of the right chiral fermions for positive q and is no longer always positive. Although the value of the potential at the location of the brane is still positive, it has a series of wells and barriers on each side, which ensures that the right chiral fermion zero mode could be trapped. Thus we may draw the following remarkable conclusion: for positive and q,the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.