Photoionization models dictate that many prominent quasar emission lines are sensitive to both the luminosity and shape of the quasars' high-energy continuum-primarily the extreme ultraviolet (EUV) and soft X-ray continuum. Unfortunately, the EUV band is severely obscured by Galactic absorption. Using data from the adjacent UV and soft X-ray bandpasses, we initiate the first large-scale, multiline investigation of correlations between the QSO soft X-ray continuum and line emission in a sample of QSOs observed by Einstein and IUE. We present a new error analysis for objective, automated line measurements, which enables us to include the information contained in weak or undetected lines. We tabulate more than 300 UV emission-line equivalent widths from IUE spectra of 85 QSOs in the atlas of Lanzetta, Turnshek, & Sandoval, then characterize the distributions of line equivalent and velocity widths (W-lambda and FWHM). We then compare these line parameters to the QSO continuum spectral energy distributions from optical through soft X-ray wavelengths, using survival analysis to incorporate any nondetections for X-ray flux and/or UV emission lines. Several correlations noted in previous studies are not reproduced here. However, we illustrate that the exclusion of undetected lines from such studies may spuriously enhance apparent correlations. We find significant correlations between W-lambda and UV luminosity (e.g., the well-studied Baldwin effect) for Ly alpha, C IV, He II, and C III]. W-lambda(C III]) and W-lambda(He II) also show previously unreported correlations with X-ray luminosity that, for C III], appears to be primary. The line ratios C III]/Ly alpha and He II/Ly alpha both show strongest dependence on l(X). W-lambda(Ly alpha) correlates strongly with spectral slopes alpha(UV) and alpha(OX) (between 2500 Angstrom and 2 keV), but not with X-ray luminosity. Using these results, we argue that one simple geometrical interpretation of the Baldwin effect (BEff) as a result of a distribution of disk inclinations is not plausible. We also provide evidence that the BEff weakens or disappears when the line emission is correctly compared to the luminosity in the continuum bandpass relevant to its production. We thus support the interpretation of the BEff as a change in spectral energy distribution with luminosity, and we predict that no BEff relative to X-ray luminosity should be found for Fe II or Mg II emission lines. Extensions of our method to samples of a wider redshift/luminosity range will be presented in a later paper, which will test these predictions.