Conditional copulas are very essential in the modeling of dependence in multivariate data in the presence of a random covariate. Several authors studied the asymptotics for the conditional empirical copula function. Bernstein polynomials provide an interesting tool for obtaining smooth versions of these non-parametric estimators. Here we provide new asymptotic results for Bernstein-based versions of estimators for a conditional copula, its first order partial derivatives and its density function. As an application we deal with the estimation of a risk ratio for bivariate data in the presence of covariate.