Should I buy a power supply for the input or nominal voltage of a DC device?
Assuming you are talking about a air pump, which often have a simple dc motor and often designed to accept car "12V" unregulated input, then you have a wide latitude of voltage you could supply. As mentioned in the comments, a typical auto supply is from 11 to 14.5V depending on the age and state of the battery, alternator, etc. 

The pump itself should not vary much in current or voltage draw, until it hits a point where it is stalling out. Like you stick a finger in the blower motor, or leave it running when the bag is full. Even then, the current draw will increase but your voltage source will stay the same and try to compensate by delivering more current until a safety feature kicks in. Current overload protection most likely, or a fuse if wired to a car outlet.

In short, go with the 12V supply, as directed.