Cubans get to stay as a carry over policy from when Castro took over. Everyone else is deemed to be leaving for economic reasons and not allowed, Cubans are deemed to be fleeing oppression. Even though other groups in Latin America have fled from civil wars, the Cubans are the only ones to get that automatic designation, all others have to prove their lives are in danger.
So, Cubans have never been part of the 'immigration' fight because of their special status. They are never deemed 'illegal immigrants' so tightening it up doesn't affect them.
They are against immigration as well because of this.