numBases = 100;  (*Number of base pairs,representing a segment of DNA*)
kappa = 1/
   10;    (*Elasticity constant,chosen to reflect the \
stiffness;arbitrary unit*)
omegaD = 
  2/10;   (*Frequency term,speculative and low to reflect slow \
dynamics*)
beta = 5/
   100;    (*Nonlinearity coefficient small to introduce mild \
nonlinearity*)

Just out of curiosity, do we need some kind of precision control on numerical computation because I see the x[n] values are very close to zero (e.g., between 2.0x10^-18 and -1.5x10^-18)?

initialGuess = 
  Table[{x[n], 5}, {n, 1, 
    numBases}]; (*or any small value close to expected solutions*)

also with this, I got nontrivial solutions

I think this now makes much more sense. This probably should be incorporated in the original top post, not only in the comment. Thanks for looking into this.

The small solution values observed in the given system, which models a DNA segment using a discretized Klein-Gordon equation, can be attributed to the interplay of various factors inherent to the physical and mathematical setup. The elasticity constant (kappa) set at 0.1 indicates a stiff system, leading to inherently small displacements in a stable configuration. Additionally, the small nonlinearity coefficient (beta) and frequency term (omegaD) contribute to the system's tendency towards smaller oscillations, reflecting the slow and mild nonlinear dynamics of the segment. The boundary conditions, fixing the ends of the segment, further constrain the amplitude of the solutions, ensuring that displacements remain minimal throughout. The choice of initial guess values near zero for the numerical solution process with FindRoot also influences the magnitude of the resulting solutions, aligning them closely with the initial estimates. Moreover, the physical parameters' scaling, chosen to represent specific properties of the DNA, naturally leads to small numerical values in the solutions, which are consistent with the expected physical behavior at the molecular or atomic level. Thus, rather than being a consequence of numerical precision issues, these small values accurately reflect the system's physical and mathematical characteristics.