Bias ply tires are best suited for riding where you need a thick stiff sidewall, both for damage protection and load bearing. Off road tires are bias ply to protect the sidewalls, both from external damage and so you can air down the tires and still safely ride without rim damage. Some big cruisers have bias ply tires, especially in the rear, because the stiff sidewall offers a lot of load support for a heavy bike.
If you aren't a dirt rider, one area that certain bias ply tires supposedly have over radials is longevity. Both the Heidenau K60 and the Mitas E07 are 50/50 bias ply tires, and there are lots of reports out there that they have extreme tread life.
If you're looking for an inexpensive tire that is good on the street but stiff safe for things like gravel or fire roads, my experience has been that the Shinko 705 (the radial version) is an ideal choice. I've used them the past couple years, and really like them, plus they're one of the least expensive (if not THE least expensive) adventure tire available.
"No matter where you go, there you are."