From the Sloan Digital Sky Survey (SDSS) Data Release 12, which covers the full Baryonic Oscillation Spectroscopic Survey (BOSS) footprint, we investigate the possible variation of the fine-structure constant over cosmological time-scales. We analyse the largest quasar sample considered so far in the literature, which contains 13 175 spectra (10 363 from SDSS-III/BOSS DR12 + 2812 from SDSS-II DR7) with redshift z < 1. We apply the emission-line method on the [O III] doublet (λλ 4960, 5008 Å) and obtain Δα/α = (0.9 ± 1.8) × 10-5 for the relative variation of the fine-structure constant. We also investigate the possible sources of systematics: misidentification of the lines, sky OH lines, H β and broad line contamination, Gaussian and Voigt fitting profiles, optimal wavelength range for the Gaussian fits, chosen polynomial order for the continuum spectrum, signal-to-noise ratio and good quality of the fits. The uncertainty of the measurement is dominated by the sky subtraction. The results presented in this work, being systematics limited, have sufficient statistics to constrain robustly the variation of the fine-structure constant in redshift bins (Δz ≈ 0.06) over the last 7.9 Gyr. In addition, we study the [Ne III] doublet (λλ 3869, 3968 Å) present in 462 quasar spectra and discuss the systematic effects on using these emission lines to constrain the fine-structure constant variation. Better constraints on Δα/α (< 10-6) using the emission-line method would be possible with high-resolution spectroscopy and large galaxy/qso surveys.