I guess if you use lynx to surf to http basic auth URLs you should switch to another browser until a security update is available. But I'm inclined to say while this is an interesting bug I don't think that's an incredibly common scenario.
Plain auth websites are pretty uncommon these days. That said, if using other authentication methods are annoying or inconvenient in Lynx (I don't know, I've never used it) then there may be some correlation between Lynx users and plain auth websites meaning this could impact a large fraction of Lynx the userbase.
This bug only affects HTTPS basic auth, given that the issue is with SNI (and, of course, that basic auth over HTTP doesn’t have any encryption to leak around).
Update: It seems there's a preliminary patch that will not fix http auth URLs, but will prevent the info leak: https://www.openwall.com/lists/oss-security/2021/08/07/7