Modal frequency (MF) prediction under varying environmental condition (EC) has received tremendous attention. Most existing methods for MF prediction, belonging to discriminative approaches, collapse when information of EC is incomplete in the prediction stage. Generative approaches attempt to construct a joint PDF of uncertain MF and EC, and then predict MF under varying EC by conducting operations of marginalization and conditioning using the joint PDF. There are two fundamental difficulties in this process. The first difficulty is joint PDF modelling of multivariate MF and EC as they are too irregular to be depicted by traditional multivariate distributions. The second difficulty is predictive PDF derivation of MF conditional on incomplete information of EC as operations of marginalization and conditioning in the high-dimensional case require huge computational cost. Aiming to resolve these two difficulties of generative approaches, this paper proposes BAyeSIan Copula-based Uncertainty Quantification (BASIC-UQ). BASIC-UQ contains three stages. Stage I introduces probabilistic model class candidates. Stage II conducts Bayesian inference on parameters and model class candidates with embedding the idea of inference functions for margins. The optimal parameters are determined in a decoupled way, and the most plausible univariate marginal PDF of each random variable (RV) along with the most plausible copula are selected. Stage III derives predictive PDF of MF conditional on incomplete information of EC. The exact solutions are derived for the multivariate Gaussian copula and the multivariate Student's t copula. Examples using simulated data and SHM data are presented to illustrate the capability of BASIC-UQ in joint PDF modelling of MF and EC, and predictive PDF derivation of MF using incomplete information of EC.