Up to DM interpretation.
Taking into account the rest of the spells effects, this could be understood in two ways (I’ve changed how the spell is written to make each way more clear):
The ground in the area is covered with slick ice, making it difficult terrain. When a creature enters the slick ice's area for the first time on a turn or starts its turn on the ice, it must make a Dexterity saving throw. On a failed save, it falls prone.
This would mean that yes, the crampons are enough to counteract the prone effect of the spell. Alternatively:
The ground in the area is covered with slick ice, making it difficult terrain. Due to the various effects of the spell (including the slick ice, battering hail, etc.), when a creature enters the spell’s area for the first time on a turn or starts its turn there, it must make a Dexterity saving throw. On a failed save, it falls prone.
This would mean protection from the slick ice alone would not be enough, and the creature would still have to save to not fall prone.
Personally, as a DM I would rule that the crampons are not enough. Having a major portion of a level 3 spell thwarted by a 2gp purchase that likely any humanoid can access seems to nullify the use of the spell. The only reason a person would have crampons already strapped to their feet is if they were expecting slippery terrain, and the value of the spell is taken away if all it does is add more slippery terrain.