There is an option in ENVI to open classified raster directly to the ArcMap. Right click on image file (not an individual band) in table of content Open File in ArcMap is the first option available. I have attached a screen shot for your reference.
In ENVI's toolbox, there is an option to convert final classified raster data into a vector
file that can be used in ArcGIS (that is if you really want to the analysis in ArcGIS).
1. In the ENVI Toolbox, select Classification --> Post Classification --> Classification to Vector. After this first step, you will have a file with .EVF extension.
2. Convert the .EVF file in step 1 to a shapefile. Using the Toolbox select, pick Vector then Classic EVF to Shapefile.