There is one solution, that helped me. You must install terminator from nightly repo. The full instruction is below.
Spoiler: after installing of new version most of your settings, which are responsible for how the terminator looks, will be changed.

1. Add the repo and update apt cache:
sudo add-apt-repository ppa:gnome-terminator/nightly-gtk3
sudo apt-get update

2. Install new version of terminator

apt-cache policy terminator

terminator:

  Installed: .98-1 500

  Candidate: 1.91-0~1779~ubuntu16.04.1

  Version table:

        1.91-0~1779~ubuntu16.04.1 500

        500 http://ppa.launchpad.net/gnome-terminator/nightly-gtk3/ubuntu xenial/main amd64 Packages

        500 http://ppa.launchpad.net/gnome-terminator/nightly-gtk3/ubuntu xenial/main i386 Packages

        100 /var/lib/dpkg/status

 *** 0.98-1 500

        500 http://ru.archive.ubuntu.com/ubuntu xenial/universe amd64 Packages

        500 http://ru.archive.ubuntu.com/ubuntu xenial/universe i386 Packages

apt-get install terminator

3. Close all terminator windows and relaunch the app

That's all. But you can see, that something changed. Colors of fonts, font, font size, color scheme, title bar name and so on.