You have too high chroma blue for CMYK. RGB blue which translates to CMYK acceptably should have saturation and brightness about 70% of max or less, exact limits depend on the used paper and print process.
Because it seems to have only blue and white, there surely exists a way to print it as is with a spot color. Consult your printer. You can agree which exact color number will be used.
As Illustrator user you have a tool for predicting the printing result. Learn how to use proof colors and get the needed color settings from your printer. Color calibrated monitor helps to get results which are accurate enough.