Without appying a voltage to a p-n junction, on both sides of the depletion region minority and majority carriers are in equilibrium, that means that there are no slopes in the carrier concentration. And without concentration slopes the carriers have no reason to wander. Remember that concentration slopes are one reason for a current, the other one is a present electric field.
no. of
electrons/holes
A
|              |d |
|nnnnnnnnnnnnnn|e |ppppppppppppp
|              |pr|
|              |le|
|              |eg|
|              |ti|
|              |io|
|pppppppppppppp|on|nnnnnnnnnnnnn
|              |n |
+--------------+--+-------------
   n-type             p-type

p-n junction polarized in forward direction

We begin applying a voltage in forward direction, that means from from p-type to n-type part.
no. of
electrons/holes
A
|              |d |
|nnnnnnnnnnnnnn|e |ppppppppppppp
|             p|pr|n
|             p|le|n
|            p |eg| n
|          pp  |ti|  nn
|       ppp    |io|    nnn
|ppppppp       |in|       nnnnnn
|              |n |
+--------------+--+-------------
   n-type             p-type

  <----------VOLTAGE------------
This means that majority carriers are being pushed towards the depletion region. When the voltage is bigger than the threshold voltage (the voltage across the depletion region), they can pass through the potential barrier. Those carriers now contribute heavily on the minority carrier concentration. Now there is a big concentration difference, so those extra minority carriers will diffuse away from the depletion region, recombining by the time.

When applying small voltages, the majority carrier concentration does not change noticeable, as there are so many of them. However, when applying large voltages, the number of majority carriers can be a limiting factor for the current.

p-n junction polarized in reverse direction

no. of
electrons/holes
A
|              |d |
|nnnnnnnnnnnnnn|e |ppppppppppppp
|              |pr|
|              |le|
|              |eg|
|              |ti|
|              |io|
|pppppppppppp  |on|  nnnnnnnnnnn
|            pp|n |nn
+--------------+--+-------------
   n-type             p-type

  ----------VOLTAGE------------>
When applying a voltage in reverse direction, minority carriers are pushed through the depletion region. However, as there are only few of them, the carrier concentration near the depletion region exhausts quickly. Thus only a very small current can flow.

Again, the majority carrier concentration does not change noticeably.

Conclusion

The current of a p-n junction biased in forward direction consists mainly of majority carriers, while in the other direction it's the minority carriers that constitute the current. Therefore it's quite clear that the forward current is so much higher than the backward current, which in normal applications can be considered zero.