If the hard drive has developed a short in the power circuitry, it's going to be as safe as the safety implementations present in your new power supply. I doubt the data connection would be a significant concern, but I would never say with 100% certainty that no bad could ever come from a defective hard drive being attached directly to a system.
Personally, if I had serious concern, I would use an
external hard drive dock to connect to the system, as this would isolate any new damage to the dock, rather than risk incidental damage to the new computer.
If your previous power supply died in a friendly manner, it should have prevented any unfriendly power output getting to the rest of the machine. This is a good example of why higher quality power supplies are always recommended.