Could I make the strength of a magnetic field oscillate up and down, by having the voltage increase and decrease while current remains the same?
I know that because of v=ldi/dt, that if I repeatedly lower and raise the dc current of an inductor at high frequencies, the magnetic field generated will be weaker, ...


Well, weaker and stronger. So far, so good. And by the way, if you repeatedly lower and raise a current, we'd tend to call it AC. The difference between AC and a varying DC is how fast you alter it compared to your timeframe of interest, so it's not a hard and fast distinction.


  ... so If i instead raised and lowered the voltage at high frequencies, would the magnet be workable for higher frequencies? 


No, the physics stays the same, at whatever frequency. Varying the current varies the magnetic field. 

But in order to vary the current, you'd need to apply a voltage to the coil. This applies at all frequencies as well.