Following the initial rumor that Arrow was getting canceled - which turned out to be true - it was then said that Supergirl could possibly get the ax and be replaced by a Superman series, with Legends of Tomorrow potentially on the chopping block as well.
Today during The CW's presentation at the Television Critics Association press tour, CW president Mark Pedowitz touched upon whether the shows could get canceled, but while he didn't confirm they will still be around, he noted regarding their cancellations: "hopefully it will be a long time from now."
"We believe in these franchises. This universe will continue for many many years," Pedowitz said about the Arrowverse.
As we see, The CW is hoping the shows stick around, but it should be said their ratings have been dwindling, particularly in the case of Supergirl and Legends of Tomorrow (see below); so if the trend continues, more than likely The CW will replace them with new DC shows. The TCA event also saw The CW confirm a new DC series will be coming to the network but stopped short of giving any clues as to what it might be.
Regarding the Supergirl cancellation rumor, the rumor offered that Crisis would create some sort of new CW DC TV universe that is said to be "drastically different." I was actually told similar information at Comic-Con in that Crisis would be used to give the entire DC Universe a soft reboot.
The rumors surrounding the Arrowverse and more possibly getting some sort of reboot makes sense as, following Crisis, Stephen Amell won't be on The CW as Arrow ends (why call it the "Arrowverse" if Stephen Amell and Arrow aren't a part of it?).
Legends of Tomorrow ratings:
(Sources: CBS, Wiki)