Model of electronic subsystem of Fabre salts (TMTTF)(2)X and Bechgaard salts (TMTSF)(2)X has been studied theoretically within Green function equation of motion approach with special attention to microscopical mechanism for antiferromagnetic order stabilization and pseudogap formation in the electronic spectrum. Conditions for pseudogap opening in the spectrum and anomalies in dispersion of current carriers are determined. Next-nearest-neighbor hopping is shown to break the perfect nesting property and destabilize antiferromagnetic state. On the basis of these results, experimental phase diagrams for quasi-one-dimensional organic superconductors (TMTSF)(2)X and (TMTTF)(2)X are interpreted.