This is an issue that really bothers me, I see it constantly. It is dangerous and bad for the horse. It puts pressure on the wither and pulls on the back. It also pulls the skin under the cinch. It can cause an ill-fitting saddle to shift to one side causing trouble for the horse when the rider tries to correct by leaning the opposite way, or worse, doesn't even notice and the saddle remains unbalanced. It can cause muscle damage, bruising to the wither, and even injure the spine.
This is an example that shows what I mean:
Before putting weight the saddle is positioned correctly
After putting weight, you will notice that the entire saddle has been pulled to the left
The most comfortable method of mounting, which applies the least pressure, is by placing the left hand on the lower neck or wither, and the right on the midsection of the seat or the far side of the cantle. Inserting the ball of the left foot into the stirrup. As you mount, you use the left hand to balance, and the left hand to pull along with hopping to gain momentum before pushing with the left leg. This ensures that the pressure is distributed evenly, and is easier on the horse, the rider and even the tack.