You can define the background color in the options for the tikzcd environment. This color will then be used for crossing over. As you did not provide a minimal working example, I use the example from the documentation of the tikz-cd package:
\documentclass[border=1mm]{standalone}\usepackage{tikz-cd}\pagecolor{yellow}\begin{document}\begin{tikzcd}[row sep=scriptsize, column sep=scriptsize, background color=yellow]& f^* E_V \arrow[dl] \arrow[rr] \arrow[dd] & & E_V \arrow[dl] \arrow[dd] \\ f^* E \arrow[rr, crossing over] \arrow[dd] & & E \\& U \arrow[dl] \arrow[rr] & & V \arrow[dl] \\ M \arrow[rr] & & N \arrow[from=uu, crossing over]\\\end{tikzcd}\end{document}Result:
