ArcGIS does not directly read or use the DPI of an image. This is because DPI has no relevance to GIS. GIS uses the pixel size in map units to determine how large or small an image will appear. DPI is only applicable when exporting for printing.
When using the rectify option, a new image is created, and at this point, DPI is not an issue. There are many 3rd party products that can convert the DPI of an image; Photoshop and the freeware application Irfanview are just two examples. Use one of these products to take the output from the rectify process and apply the appropriate DPI.