I'm a little late to the party, but I wanted to get my idea out there. Since Supergirl started on TV, there have been rumors of the show crossing over with The Flash. I think this is a great idea, mainly because I like things to be connected, and so far the Arrowverse consists of Arrow, The Flash, Constantine, Vixen, and Legends of Tomorrow. So why not one more?
Ever since Supergirl was announced, and Greg Berlanti was making it, I automatically put that show in the Arrowverse. It fits very well, and there are many ways I've come up with for it to crossover.
My best idea, is for them to just have a normal crossover. Maybe a metahuman is causing a problem in National City, and The Flash, Vibe/Cisco, and Caitlyn go there to capture the metahuman. In National City, Supergirl tries to fight the metahuman, and eventually the Flash gets there and helps. Maybe Cisco and Caitlyn stay with Winn and James to try and help take down the metahuman, and the Flash and Supergirl work with the DEO to take him down. I think this is the best idea for a crossover, and it sets up the potential for Supergirl to crossover with The Flash again, Arrow, or Legends of Tomorrow.
Another idea I have that I really don't like, but is possible, is to have Supergirl set on another Earth. In season 2 of The Flash, the concept of different Earths was brought into the show. So what if Supergirl existed on Earth-2, where Jay Garrick was from, or another Earth? I think this may be the most likely to happen, but it's a very bad idea. Like Arrow star Stephen Amell said, "Any DC TV show can crossover." While I don't completely agree with him, I like most DC TV shows existing in one universe, all on the same planet.
I also have an idea for a crossover that would be very subtle, but it would connect Supergirl to the Arrowverse. On Supergirl, Kara killed Red Tornado and Alex killed T.O. Morrow. I am not totally convinced Red Tornado is dead, or I think he could come back. Maxwell Lord does have Red Tornado's arm, and while he is using it for a different purpose, he could be creating a new version of Red Tornado. What if he does end up creating a new version of Red Tornado, but the android doesn't obey, and leaves National City. After that, the Legends of Tomorrow could recruit him and he can become a member of the team.
My last idea for a crossover, is Martian Manhunter. What if in the future of Supergirl the story does not require Hank Henshaw? Because of that, the character could move to Legends of Tomorrow, and be a member. That would be a very interesting addition to Legends, with Hank being a Martian and all... so that show would have a bunch of humans from present day, a time-traveler from the future, and an alien from Mars.
There, I said it. I want Supergirl apart of the Arrowverse. And I don't want them to be connected because Supergirl is set on another Earth. All of the shows work very well together, and they all have the same people behind the scenes, like Greg Berlanti. This show needs to be connected, because to me, most DC TV shows are already connected, and I'd like it to be official.