The main reason why Basque people have been excluded is nationalism. The nationalism is both on their part and on the part of the people around them.
While Spain, in particular, is a multi-ethnic country to some degree, the Basques are much more different than the other ethnic groups in Spain. The clearest example of this is their language. While Catalan and Castilian, for example, are somewhat different, Basque is completely different. Basque is not even in the same family as those other languages. It is not an Indo-European language at all. Thus, the Basques are ethnically very different from the other groups in Spain and in France.
Because of their differences, there is some difficulty in getting them and the other people to cooperate with one another. We do not know for certain why nationalism exists and why it becomes important in some places and not in others. However, we do know that it exists and is important. Because the Basques felt national pride, they did not want to assimilate with the French and Spanish. Because the French and Spanish felt national pride, they wanted to force the Basques to assimilate. When Basques do not wish to assimilate, they end up being excluded.