Did the Portuguese Crown sponsor Brazilian gold-seeking expeditions (bandeiras) in the 17th century?

After a fashion, yes.

The {Brazilian gold rush began at the end of the 17th century in the mountainous province of Minas Gerais (General Mines in English), a rich ore-producing region to this day. Perhaps a million people, close to half of the population of Brazil "went south" (Brazil's "west") from the Atlantic coast in the northeastern part of the country. Such a migration would be followed with great interest by the central government. Indeed, the Portuguese government sent a large number of "administrators" to these areas to track, regulate, and tax the gold production, albeit with a time lag to actual developments.

Now, if the question is were gold-seeking expeditions "fully" sponsored by the Portuguese crown, the answer is probably no. That is, local explorers with knowledge of "local" conditions further out on the edge of the country (e.g. near the Paraguayan border) pursued their own exploration activities without the sponsorship of the government. It seems like the "government" was always at least one step behind.

About the context of the bandeiras (unsponsored expeditions) from S. Paulo (formerly S. Vicente captaincy)

In short: In much of the highlands, specially S. Paulo, outside of the sugar cane coastal regions, the dominant language was Tupi until the Portuguese intervened, and the Indians were a common race in the racial mix, up to the arrival of black slaves and white gold rushers to Minas Gerais in the XVIII c., and later, of new European immigrants in the XIX c. The paulistas (people from S. Paulo or S. Vicente captaincies) were backward and economically disconnected from the colonial economy based on sugar cane or commerce with Asia/Europe/Africa, and thus enslaving Indians or looking for gold must have appeared profitable. They needed not to be sponsored by the Crown, they were desperate and/or independently minded enough.

The following is a short story about how S. Paulo state was settled by a mix of Indians and Portuguese with Jesuit help:

Even today Brazil's census show ~30% white, ~30% black, and ~40% "Pardo" population. "Pardo" literally means an intermediate, undefined color, the word may be used for a grey or brown — but it is only used for living things (a cat may be pardo). Many pardos may be mulato (black/white mix), but it is impossible to know how many of them (or how many of the self-declared as white or black) are partly Indian.

Portuguese married (or just had children) with Indians since the XVI c.. Statistics such as "1% Native American" are true but misleading, since 1% counts only the remaining "pure" Indians, who were isolated in more remote regions up to more recent centuries (or up to now). Many Indians were absorbed in the general population.

João Ramalho was exiled in the early XVI c. S. Paulo coast as punishment for some crime. He, alone, got friendly with the Tupi. He would travel a lot between different Tupi tribes because the Indians were anxious to know something about the new strange visitors, and, as the Tupi would easily offer their young girls to friendly visitors, he got a lot of children.

When, a generation after, the Portuguese fought the French, an older João convinced his Indian friends to join the war and his many children were leaders of the allied Indian troops. It helped that the French allied themselves with their ancient tribal archenemies.

These Indians also helped to populate many new cities: São Paulo was founded by a Jesuit mission among the tribes friendly to João Ramalho, and many people bragged to be his descendants afterwards (BTW, Wikipedia says that the current Queen of Sweden is his descendant — I did not know).

In São Paulo, Tupi was spoken and even written and taught (the Jesuits devised a written form for the language), up to 1750. Then the Portuguese invalidated legal evidence in languages other than Portuguese as they were tired of seeing local legal processes and inheritance depending on local documents / witnesses in Tupi — it was too difficult for Portuguese Courts to serve as second instance courts in this situation. They also prohibited to teach Tupi at schools. Up to 1850, it was still possible to find old people whose first language was Tupi.

Many place names in Brazil have Tupi origins and some of them are not ancient Indian names (e.g., Bertioga) but were new cities named in Tupi by the paulistas themselves (e.g., Sorocaba, founded in 1654).

The state of São Paulo and much of the Brazilian hinterland was out of the sugar cane cycle, as it was too hard to move produce from the highlands (S. Paulo city is ~750m above sea level) to the coast (sugar is heavy). An exception was the small São Vicente coast in Sao Paulo, but this was small compared with the northeast sugar area. So, at the beginning of colonization, in the highlands there were no white plantation masters and black slaves.

Besides that, unlike the Indian languages in North America, Tupi dialects were spoken widely, from what is now Paraguay to large parts of what is now Brazil. Thus, the work of the Jesuits to develop a written form and standardize the language resulted in a lingua franca, called 'língua geral' which was pretty useful for inter-regional trade and helped the Bandeirantes to settle and trade far from their homes.

The Bandeirantes, who went out of S. Paulo captaincy to the deep hinterland — or even to Paraguay and the Jesuit missions, to enslave Indians — were themselves of mixed race, although they had Portuguese names. When some of them were hired to fight a slave rebellion in the more developed/civilized Brazilian Northeast, the bishop even complained that "these southern barbarians do not even know how to speak" — i.e., they could not speak Portuguese. The Northeast was actually more developed (sugarcane + African slaves) up to the 1700s, when gold was found in Minas Gerais. Then, in the 1800s, when coffee started to become important, S. Paulo also become important.