It is suggested that the use of "trooper" was intended to evoke the image of the Australian mounted State police. A trooper, in the army, is a member of a cavalry troop, a soldier on horseback.
The introduction of state police in the USA was controversial, with the unions believing that they would be a military-style gendarmerie, who would be brought in to break up strikes and other union actions. The use of "trooper" suggested the Australian Troopers, who were generally better accepted by communities that they policed. The original Australian troopers were mostly British cavalry ex-soldiers, and so were "troopers" in the original sense.
Moreover, the first state forces rode horses and even now, patrolling the state highways on motorcycles is a particular role for the various state police forces. The image of a trooper on horseback easily transferred to that of a trooper on a motorcycle, so in this sense, the name is appropriate.