Is it safe to use a new microwave if I dropped it on the back corner?
I'm not sure whether or not this question is really on-topic here.  But the basic answer is that microwaves generally have a lot of safeguards built in to avoid radiation leakage, if that's what you're worried about.  The FDA states:


  There is little cause for concern about excess microwaves leaking from
  ovens unless the door hinges, latch, or seals are damaged. In FDA's
  experience, most ovens tested show little or no detectable microwave
  leakage. If there is some problem and you believe your oven might be
  leaking excessive microwaves, contact the oven manufacturer, a
  microwave oven service organization, your state health department, or
  the nearest FDA office.


So, if only very minor exterior damage was done, and there was no damage to the door or seals, it seems unlikely there would be a problem.

On the other hand, WHO recommends:


  The design of microwave ovens ensures that the microwaves are
  contained within the oven and can only be present when the oven is
  switched on and the door is shut. Leakage around and through the glass
  door is limited by design to a level well below that recommended by
  international standards. However, microwave leakage could still occur
  around damaged, dirty or modified microwave ovens. It is therefore
  important that the oven is maintained in good condition. Users should
  check that the door closes properly and that the safety interlock
  devices, fitted to the door to prevent microwaves from being generated
  while it is open, work correctly. The door seals should be kept clean
  and there should be no visible signs of damage to the seals or the
  outer casing of the oven. If any faults are found or parts of the oven
  are damaged, it should not be used until it has been repaired by an
  appropriately qualified service engineer.


Basically, I can't know how much damage and/or what type of damage you did by dropping it.  (And no one else here could know for certain either.)  Microwaves are generally designed in such a way to prevent radiation exposure except in cases of severe damage or deliberate disassembling, but I have no idea what may have happened internally to your device by dropping it.

If you're concerned about other safety issues (electrical shorts, etc.), it would be even more speculative to try to address whether or not they could happen with these few details.