This paper studies a nonlocal boundary value problem with Steklov’s conditions of the first type for a linear ordinary delay differential equation of a fractional order with constant coefficients. The Green’s function of the problem with its properties is found. The solution to the problem is obtained explicitly in terms of the Green’s function. A condition for the unique solvability of the problem is found, as well as the conditions under which the solvability condition is satisfied. The existence and uniqueness theorem is proved using the representation of the Green’s function and its properties, as well as the representation of the fundamental solution to the equation and its properties. The question of eigenvalues is investigated. The theorem on the finiteness of the number of eigenvalues is proved using the notation of the solution in terms of the generalized Wright function, as well as the asymptotic properties of the generalized Wright function as λ → ∞ and λ → −∞.