There were four sets of recordings that came out of those sessions, and they had a significant influence on things that came out over the next few years.
First up, a selection of recordings from those sessions circulated around Dylan's music publishers and Artist and Repertoire departments of the major music companies. Dylan was picking up significant royalties from covers of his songs so it made sense to get some of his new material out into the marketplace.
That demo tape yielded, among other hits, Manfred Mann's The Mighty Quinn, The Byrds’ You Ain't Goin' Nowhere, Peter Paul & Mary's Too Much of Nothing and Julie Driscoll's This Wheel's On Fire (with, of course, the Brian Auger Trinity).
The tape itself, however, was a small subset of a much broader collection of covers and trad. arr. tracks that provided the basis of some of the earliest manifestations of the bootleg music industry.
After the Basement sessions, Dylan's next studio album, John Wesley Harding, a collection of material that hadn't appeared on those tapes, marked a pronounced change from what had gone before and it was followed by a different album that changed the musical directions of, among others George Harrison, Eric Clapton and Fairport Convention, producing, as a result, the Los Angeles band we know as Los Lobos.
The news that Dylan's backing band was working on the album we now know as Music From Big Pink wasn't something that produced bated breath anticipation, however. My mate Eric, for instance, dismissed the news with the comment that I'm not interested in Dylan's backing band, I'm interested in Dylan.
Ah, the rash innocence of youth.
The Basement sessions, while they provided some of the material that turned up on Big Pink, had an additional significance to the unit that became known as The Band. They also provide an interesting insight into the ongoing enigma that is Bob Dylan.