I have a set of equivalent width measurements for different emission lines. How can I use them in order to determine the corresponding line luminosities?
If you also have photometry, then there is a reasonably accurate conversion between the apparent magnitude and the continuum flux (per unit wavelength interval) at the wavelength of the photometry.
Once you have this conversion factor, you multiply your equivalent width by it to get a flux. To turn that into a luminosity then requires a distance and luminosity is flux multiplied by $4\pi d^2$, assuming there is no extinction.