In our recent work [Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], a family of high-order asymptotic preserving (AP) methods, termed IMEX-LDG methods, are designed to solve some linear kinetic transport equations, including the one-group transport equation in slab geometry and the telegraph equation, in a diffusive scaling. As the Knudsen number goes to zero, the limiting schemes are implicit discretizations to the limiting diffusive equation. Both Fourier analysis and numerical experiments imply the methods are unconditionally stable in the diffusive regime when . In this paper, we develop an energy approach to establish the numerical stability of the IMEX1-LDG method, the subfamily of the methods that is first-order accurate in time and arbitrary order in space, for the model with general material properties. Our analysis is the first to simultaneously confirm unconditional stability when and the uniform stability property with respect to . To capture the unconditional stability, we propose a novel discrete energy and explore various stabilization mechanisms of the method and their relative contributions in different regimes. A general form of the weight function, introduced to obtain the unconditional stability for , is also for the first time considered in such stability analysis. Based on uniform stability, a rigorous asymptotic analysis is then carried out to show the AP property.