Usually the insides of drawers are unpainted, but if it is rough or unsightly you can paint them. I wouldn't use a stain but a paint. Do it to suit yourself and what you want to store in them. After all they are unseen until you pull out the drawer

thanks oilpainter, after doing some searching for painted dressers, it appears even the most spendy ones do not have the sides painted, and the drawers are usually lined with some decorative paper.