Interacting bosons generically form a superfluid state. In the presence of disorder it can get converted into a compressible Bose glass state. We study such transition in one dimension at moderate interaction using bosonization and renormalization group techniques. We derive the two-loop scaling equations and discuss the phase diagram. The correlation functions at the transition are characterized by universal exponents in a finite region around the fixed point. We apply the derived theory to an XXZ spin-1/2 chain in the presence of several types of disorder that do not break the XY symmetry and calculate the complete asymptotic form of the spin-correlation functions at zero temperature at the transition between the liquid and the disordered phase. Apart from a universal power law decay of correlations, we find additional logarithmic corrections due to marginally irrelevant operator of disorder.