The basques and galicians, as subjects of the Crown of Castile, particpated in the american colonization, and very activelly by the way, specially since the latter 17th century, but also before. On the other hand the main sources of colonists in early stages of the colonial expansion in the americas were extremenian and, mainly, andalusians. I may be wrong, but I think there is an (understable) trend amongst anglo-saxons to mirror spanish diversity wih british diversity, an Castile with England, but things doesn't work in the same way in both places.
In this case, you have to diferenciate the Crown of Castille and Leon and the kingdom of Castille (or kingdoms, since there were two), because the former was already a multi-cultural and multi-lingual entity. So, when we speak about "castilians" in this context it can be confussing, because in general terms basques, galicians, andalusinas etc were castilians, but in a more specific sense they weren't (and aren't), in the same way that not all the subjects of the Crown of Aragon were aragonese. Though I doubt it was a big issue back in the time.
Castille properly, occuping mainly the central iberian plateau, had a rather low population.
On the other hand, the reasons behind the geographical origin of the colonists is more related, well, with geography and the emplacement of the main ports than with any other consideration. First Cadiz/Seville, and latter La Coruña became the main (and legal) gates towards the Americas. There is also other demographical and economical considerations, but well, we can pass without mentioning them for the moment. Thus for a Catalan, a Valencian etc it was easier to go to look for fortune in Italy or north Africa than making a long and incertain voyage to Seville or La Coruña with not much incetives for them.
Still, there are documented catalans, valencians and aragonese since the early stages of the colonial expansion, some of them in very important political positions, and probably many others that didn't let historical traits also participated (difficult to know in a time without ID cards), though, for the reasons expalined above, in small numbers. Without looking too much, the monarch's military representative in Columbu's second voyage, Pere de Margerit, was catalan as many of his men and pretty much they conquered Hispaniola. It was trade what wasn't allowed by no castilians (I mean non-subjects of the castilian crown), not settlement, by the way (and again, with the means of the time it would have been virtually impossible to apply this banning).
Also latter, after the Nueva Planta decrees that allowed aragonese trade with the Americas, many subjects of the aragonese crown, specially catalans, settled in the americas, though this procecess is more intense in the 19th century in the remaining spanish possesions in the Caribbean . Do you know Bacardí rum? well, thats a very catalan familly name, for example.
So, in order to have less castillian "domination" in the colonization, you have to move the aragonese kingdoms in the map. I can't see another form. If its about. Still, something that could attract bigger numbers of aragonese subjects to the Americas would be if they had been allowed to trade with the Americas since the beggining, though the demographical wheight of the castillian crown would be still decisive. On the other hand this would be in contradiction with the poli-synodial system followed by the Habsburgs, so perhaps we need a different dinasty.