Adobe Illustrator wont open on Sierra
Was this a clean install of Adobe or an "restore" from a Time Machine backup?  I have personally had issues when going from one issue to the next (Mavericks to Yosemite and Yosemite to El Capitan) and migrating my software instead of clean installing it.  Doing a clean install fixed my issue each time.

So, first thing to do, is uninstall and do a clean installation of Illustrator.

At the time of this writing, Adobe wasn't experiencing any issues with Illustrator.  See macOS Sierra (10.12) compatibility FAQ | Creative Cloud

However, they do provide a link for people to report bugs.  If doing a clean install doesn't solve the problem I would report the problem.