How can one argue against climate change deniers and antivaxxer and other extreme and potentially harmful anti-science views, given the Duhem-Quine thesis, theory ladeness of facts, and the underdetermination of theories by data?
On the practical side of things, in many (most) cases you actually can't have an effective rational argument in real time against some positions. Many salient beliefs are not held on a strictly rational basis, and not easily/directly changed by rational argumentation. The necessary existence of auxiliary hypotheses isn't the cause of intransigent beliefs, instead it provides the levers by which people can maintain intransigence in the face of rational argumentation. Sometimes, taking a more oblique approach, e.g. addressing underlying concerns like "government takeover of the economy" for climate change or concerns about "big pharma" for vaccines"big pharma" for vaccines, might be a way to go.
One thing to keep in mind is that "everyone who believes the truth, believes the same thing, each person who believes a falsehood believes slightly differently.", so you need to pay attention to the specifics of the overt beliefs on the topic, and the underlying world view in order to communicate effectively.
This aphorism also bears on the (rare) case where you actually are dealing with people who are willing/able to enter into a rational discussion. Then there is the practical difficulty of being able to back up any/all of the auxiliary hypotheses (of which most (almost all?) are backed up by evidence) that your interlocutor may object to. Alice rejects ice cores as a viable record of climatological data? Get your geochemistry hat on. Bob likes the brighter sun theory? Get your astronomer hat on etc. If you are versed in these areas, and your partner is in a mental state to be swayed by rational argumentation, then you might be able to get them to accept the underlying assumptions. If you and they accept the same auxiliary hypotheses, then you'll have to agree on the overall conclusions. In our technologically advanced, and specialized, world it would take years of study to really understands the ins and outs of the actual scientific consensus in a way that can be effectively deployed to try and convince people on these controversial issues.