What is the preferred method for exporting and printing items in ArcMap?

Answer

It is highly recommended to use ArcGIS Pro for printing and exporting, especially when experiencing issues caused by the limitations of the ArcMap display engine. More specifically, ArcGIS Pro is not restricted by the graphical device interface (GDI) limitations that some users experience in ArcMap. For example, transparency is natively supported in ArcGIS Pro, preventing the rasterization of layers. Additionally, ArcGIS Pro supports transparency in layout elements.

Note:
ArcGIS Pro does not support exporting directly to AI (Adobe Illustrator) files as it is a closed file format.  
It is recommended to export to the AIX format and use the ArcGIS Maps For Adobe Creative Cloud extension in Adobe Illustrator to open the exported AIX files.  
For more information on the subtle differences between ArcMap and ArcGIS Pro in drawing, please see the help topic: Visual differences in imported map documents