Alison Green from Ask a Manager had this to say about the use of LinkedIn:
I can’t imagine rejecting an applicant because she wasn’t on LinkedIn, unless the position I was hiring for included social media, in which case it might raise my eyebrows. I think that recruiter really overstated this to you.
That said, it’s true that the vast majority of applicants for professional jobs these days do have LinkedIn profiles, and if I notice that someone doesn’t, it does feel a little off. Not like “this person is horribly flawed,” but more like, “huh, I wonder if she’s less keyed into professional trends.”
It’s not going to stop me from hiring someone … but there’s also no reason not to just throw together a profile and have some reasonable presence there. Being on LinkedIn doesn’t really require more than setting up a profile; you don’t need to be constantly logging on or anything like that.
I’d think of this kind of like sending a thank-you note: Neither is likely to be a deciding factor on your candidacy, but both take minimal time and make you look more polished, so it’s hard to find a compelling reason not to do it.
Source: do I really need to be on LinkedIn?, 2015-03-31
So in short: not having a LinkedIn profile isn't a red flag unless social media are a key element of your work, but it's better to create a simple profile to avoid looking out of touch with modern workplace trends.
Speaking from experience, not having a profile would be even stranger for certain "modern" fields like IT and certain industries like consultancy. Consultants may even be required by their employer to maintain an up-to-date LinkedIn profile.