Heat 2 tablespoons oil over low heat in a 12-inch non-stick skillet (so potatoes don't stick). While skillet heats, prepare onion and crab as directed above. A few minutes before cooking, increase heat to medium-high. When oil starts to send up wisps of smoke, add onion and crab; cook, stirring often, until golden brown, 4 to 6 minutes. Meanwhile, dice potatoes and toss with remaining oil. Transfer crab mixture to a bowl and reserve.
Add potatoes to empty skillet; cook, stirring only occasionally so they form a golden-brown crust, about 10 minutes. As potatoes cook, mix ketchup, mustard, Old Bay™ Seasoning, fresh basil and 2 tablespoons of water. (Recipe can be prepared to this point up to 2 hours ahead. Spread hot potatoes on a large lipped cookie sheet; cover when cool. Return skillet to medium-high; add potatoes and re-crisp.)
Return reserved crab mixture to skillet; stir in ketchup mixture, then season with salt and pepper. Cook, stirring frequently, until hash has nicely browned, about 5 minutes longer.