Batwoman has confirmed that the Joker exists in the Arrowverse. The Arrowverse has long tended to use ideas that are traditionally associated with Batman; indeed, Arrow's version of Oliver Queen is basically a Batman who's taken archery lessons. Prominent villains have included the likes of Ra's al Ghul, and Arrow even began to develop its own version of Suicide Squad for a while, drawing members from Batman's rogues' gallery.
Batwoman, of course, takes things a step further. The show is based in Gotham City, which has apparently been abandoned by Batman for the last three years. Last year's "Elseworlds" crossover event introduced a number of classic Batman villains, and gave explicit nods to others; Arkham Asylum was confirmed to be home to the likes of the Penguin, Poison Ivy, Clayface, the Riddler, and the Scarecrow.
The Batwoman season 1 premiere has now confirmed that the Joker exists in the Arrowverse as well. In fact, it seems the Clown Prince of Crime is involved in Kate Kane's revised origin story; Kate's mother and sister were seemingly killed in a car accident that was triggered by the Joker. Batman's attention was divided between saving the civilians and catching the Joker; he thought he'd achieved both by grappling the car carrying the Kane family, and headed off in pursuit of his nemesis. Unfortunately, he'd failed to factor in bad engineering, and the car dropped into the river.
It makes sense for the Joker to exist as part of the Arrowverse; Harley Quinn was referenced in a couple of episodes of Arrow, and it's pretty hard to imagine a scenario where Harley exists but the Joker doesn't. But, in addition to confirming the Joker's existence, this flashback reveals the nature of the dynamic between Batman and the Arrowverse's Joker. Batman was pursuing the Joker back when Kate Kane was still a child, meaning both he and the Joker have been active in Gotham for over a decade. The Arrowverse's Batman, then, is a skilled and experienced vigilante, who protected Gotham for years before his recent mysterious disappearance.
It's generally assumed that Batwoman will find herself crossing paths with a number of classic Batman foes, including the threat of Hush; it will be interesting to see whether or not the Joker is one of them. DC has traditionally been wary of developing characters on both the big screen and the small; that's why Arrow pivoted away from Suicide Squad. But there have been recent signs that may be changing since the Warner Bros./AT&T merger, so it can't be ruled out.