In the grand scheme of things I'd probably liken it to disability ramps, braille on room signage, etc. It'll probably add a few points to the total cost, but in the grand scheme we're talking about 3-5%, not 30-50% in additional costs. What'll probably happen is that each prof will need to assign a TA to basically deal with it. (Remember, these specialized technical costs are particularly cheap for universities.)
I unfortunately don't have a lot of insight on the case itself, but I surmise the reason why these activist lawyers are going after Harvard and MIT is because they have plenty of resources to solve these issues. Part of being a leading academic institution is to "do the right thing", and at this juncture it's up to highly specialized legal people to figure out what that means.
Almost all TAs would take forever to caption video and would do a bad job. TAs don't grow on trees and transcribing captions for all the lectures would use up their full 20hrs/week. A lot of the cited videos in the suit are not of course lectures but from events on campus. Good commercial transcription service costs about ~$150/hour with markups for difficult video (bad audio quality, accents, obscure subject vocabulary) but an institution doing a lot of business will get a discount.
An advantage of bringing the suit is it brings in the federal government as "referee." The advocates have their ideas about what "the right thing" is and Harvard administrators and lawyers will have their own but the feds will basically create regulation for how these long-standing laws should be applied in these cases. They might say that certain kinds of content should be captioned as a matter of course but that others can be left uncaptioned until an individual requests it, the analogy being that the school doesn't have to have an ASL interpreter at every on-campus event, just at the ones where one is requested.
I unfortunately don't have a lot of insight on the case itself, but I surmise the reason why these activist lawyers are going after Harvard and MIT is because they have plenty of resources to solve these issues. Part of being a leading academic institution is to "do the right thing", and at this juncture it's up to highly specialized legal people to figure out what that means.