Given examples elsewhere, might Barclays Center suites and private spaces get used while the arena's not open?

With a community proposal to use the Barclays Center for public school uses and an agreed-on NBA plan to use arenas, including Barclays, for voting--the atrium, at least--it's worth a look at an 8/24/20 Sportico article, SOFI STADIUM BUSINESS SUITES HINT AT THE FUTURE OF MULTI-USE BUILDINGS:

As teams get more creative selling access to their increasingly expensive stadiums, SoFi Stadium [in Los Angeles] is uniquely promoting a benefit for its suite owners: year-round access to their luxury boxes.
In a rare if not unprecedented offer for a major American sporting venue, therams.com mentions holding business meetings and private parties as potential off-day uses for their boxes, which can come with “365-day access to SoFi Stadium.” 

Perhaps more on point is access at Talking Stick Resort Arena in Phoneix, where "Suns global partner PayPal has special access to its corner box even when the building is empty, where it has created a so-called 'innovation center.'"

In Brooklyn, that could mean access, via the VIP entrance on Atlantic Avenue, to suites and clubs, perhaps even early in the arena's reopening phases. Presumably the number of people, as well as sanitation safeguards, would be easier to manage for a relatively small number of suiteholders.

That said, it's questionable whether (or how much) even that would come back before there's a vaccine. But even if a vaccine emerges, some people will be skittish about gathering, so it would not be surprising for venue managers to cater to high-end clients.