Well, the Daynes’ prominence comes primarily from the fact that A. they’re one of the oldest Houses in Westeros, which gives them a lot of prestige, B. they’re also former royals, which gives them additional prestige, and C. they have a famous warrior tradition of the Sword of the Morning, which again gives them quite a bit of prestige in a warrior culture.Thus, despite not being one of the richest or most powerful of the Lesser Houses of Dorne, they have a ton of social/cultural capital.
In terms of their loyalty, I wouldn’t call the Daynes particularly notable as Targaryen loyalists beyond Ser Arthur Dayne himself – we don’t hear about any Daynes fighting at the Battle of the Trident, for example, nor do we see any Daynes of Starfall numbered among Oberyn’s partisans or Arianne’s conspiracy, and they arranged a betrothal to one of Robert Baratheon’s bannermen.