Quine did not specifically study "higher set theory", and his positions on the issue are mostly generalities following from his empirical holism (mathematics is the "entrenched" part of the "web of belief" that touches on experience at the observational boundaries) combined with the indispensability argument (what is indispensable in empirical science, e.g. sets and numbers, must be granted ontological rights). His more detailed remarks on the issue came in response to Parsons' critique of his positions in Review of Parsons (1984) and Reply to Parsons (1986):
"So much of mathematics as is wanted for use in empirical science is for me on
a par with the rest of science. Transfinite ramifications are on the same footing insofar as they come of a simplificatory rounding out, but anything further is on a par with uninterpreted systems." [Quine 1984, p. 788]
"I recognize indenumerable infinities only because they are forced on me by
the simplest known systematizations of more welcome matters. Magnitudes in
excess of such demands, e.g., ב_ω [the cardinal number of V_ω(N) and of V_ω+ω] or inaccessible numbers, I look upon only as mathematical recreation and without
ontological rights." [Quine 1986, p. 400]
Quine is even more explicit in Pursuit of Truth, where he comes out in support of the axiom of constructibility due to "the considerations of simplicity, economy, and naturalness that contribute to the molding of scientific theories generally", and because "it inactivates the more gratuitous flights of higher set theory, and incidentally it implies the axiom of choice and the continuum hypothesis", thus "rounding out" ZFC. This is followed by a favorable mention of constructivism and predicativism, which of course do an even better job of "inactivating" the higher set theory. Indeed, Field's fictionalist programme of reducing mathematics of natural sciences to a predicative minimum is close to the realization of Quine's own youthful dream (with Goodman) of complete nominalism about mathematics. If successful, it would allow to dispense with the indispensability argument, and platonism even about numbers. See Burgess's Why I Am Not a Nominalist, who calls Quine and Putnam "false friends of numbers" on this score.
As mentioned, Parsons pointed some incongruities in Quine's views of mathematics. A detailed critical analysis of this critique and Quine's responses to it are in Tieszen's book chapter Gödel and Quine on Meaning and Mathematics:
"Parsons notes that, according to Quine, there is no higher necessity than physical or natural necessity. Set theory is supposed to be on par with
physics in this respect. The problem is that there is tension in Quine’s
own view of this. It conflicts with his view of mathematical existence...
Quine is a platonist about set theory. The notion of object in set theory, however, and the structures whose possibility it postulates, are much more general than the notion of physical object and spatiotemporally or physically representable structure. But then how can Quine maintain that these possibilities are ‘natural’ and that the necessity of logic and mathematics is not ‘higher’?
[...] This is related to another difference noted by Parsons: elementary
mathematical truths do not seem to be even more rarefied and theoretical
than the theoretical hypotheses of natural science. On the contrary,
they seem quite obvious. Quine’s view cannot explain the obviousness of elementary mathematics and parts of logic (Parsons 1980, p. 151). In fact,
there seem to be very general principles that are universally regarded as
obvious, whereas on an empiricist view one would expect them to be bold
hypotheses about which a prudent scientist would maintain reserve."