Little or not at all.
By now everyone who reads DC comics knows Carrie Kelley, Robin-of-the-future in Frank Miller's famous The Dark Knight Returns (hyperbole! Rolling Stone magazine!), will make her in-continuity debut in Batman and Red Robin #19. The cover is a Miller homage complete with lightning bolt, but the interior pages DC's shown are more of a clever goof on his version of Carrie. And maybe Dan Clowes as well-- she looks like a cosplaying Enid Coleslaw. This Carrie is older than Miller's and kind of out of her timeframe, plus it doesn't seem she'll be an actual Robin.
While the continuity-minded might balk at introducing a character from the future when so many perfectly good characters from the current day don't seem to exist as yet-- for example, Lady Shiva hasn't given birth to Cass Cain-- but I've always liked Carrie. And the New 52 continuity doesn't have to proceed in the same way chronologically as the classic DC continuity. In fact, it's pretty obvious it hasn't.
The original Carrie Kelley was a fun Robin, the daughter of two neglectful parents who goes a-heroin' with a grizzled, middle-aged lunatic who's about as warm as a penguin's ass. Miller's teen girl dialogue is frequently groan-inducing-- probably from hanging out with Chris Claremont too much-- and Carrie's survival strains suspension of disbelief in a story that's at its breaking point almost from page one. But she won me over in the end as a bright contrast to the gloom and doom. But for a number of reasons that should be obvious, this Carrie isn't that Carrie. She may have some of the original's likability, but I find it unlikely she's going to be a true Robin.
She'd need lots of training, unless she just happens to be a highly-skilled martial artist. Which I doubt. Sure, they might be hinting at that by giving her a Damian connection of some sort, but Batman coming across a fully qualified rookie Robin isn't exactly what series writer Peter Tomasi calls "organic and heartfelt." More like serendipitous and shoe-horned. Unless he's making some kind of wicked joke on some already very sensitive fans. I refuse to believe in conspiracies and take the man at his word. DC has made some boneheaded decisions lately but I doubt anyone there is that cruel and/or dumb.
And making Carrie a true Robin would be cutting in line, right? If it's to be a female Robin, there's already a likely candidate in Harper Row who logically should have a shot. From what I've gleaned online, I've come to think they're positioning Harper for that role anyway.
While we're on that topic, I favor there being a female Robin. While it's already been done at least twice, it's worth going there again. Having a female Robin makes for a different kind of mentor-protege relationship between the two heroes and that deserves further exploration because it offers new story possibilities. And, after all, how many short, black-haired boys with acrobatic skills can there be in Gotham City?
Anyway, getting back to Carrie's affect on the continued non-existence of Cassandra Cain. I don't really see how the two relate.
Over at the Mary Sue, a few commenters have registered some unhappiness with Carrie's appearance. Mostly having to do with DC's continued refusal to use Steph and Cass in stories:
"And for HER, we get cheated out of Steph and Cassandra? AGAIN???"
"Cassandra Cain would slide nicely into Damian's role."
One person totally disagrees with me and rejects the idea of a female Robin (fair enough). Another writes he's more annoyed with Cass and Steph not being Batgirl again than with the possibility either of them have been replaced as potential Robins.
The last I tend to agree with a bit more than any of the others I read there. Cass has never been identified with the Robin role in the way Steph has. She was always Batgirl, the first to have her own solo title. Plus, I think of Robin as a lower ranking title. Batgirl has more autonomy (like the fully adult Nightwing before her, Cass received an entire city of her own to patrol and clean), whereas Robin is always a kid or teen apprentice of some kind. Why would Cass settle for being a Robin when there's no one short of Batman himself she can't beat in a fair fight? Also consider that Cass another well-established super identities of her own-- Blackbat/Black Bat. If we must have the original Batgirl in the costume, and given the plethora of Robins, there's really no reason we couldn't have had Cass as Black Bat-- at least until they de-aged Lady Shiva and gave us the in-story reason for it. But as Robin? No thanks. Even Red Robin would be more appropriate, and even that costume doesn't carry the status of Batgirl's, or offer the potential for creating her own legacy as does Blackbat's.
So introducing some college kid and having her wear a Robin costume at a party in no way impinges on Cassandra.
The "but Steph" arguments carry more weight with me. Still since there's no indication Carrie will be a ROBIN-Robin, and given that Harper Row fulfills a lot more of Steph's functions in the Bat-family, and considering Steph's promotion to Batgirl, it seems unlikely Carrie is an either-or substitute for Steph. I really doubt anyone at DC ever made the connection. Now if she were wearing a Spoiler costume, then I could definitely see cause for major alarm.