Seems silly, but if you guys have the same problem as me (fonts are way too huge for everything to fit inside the unsizeable launcher window), click anywhere inside the launcher and drag, as if to highlight, towards the right/bottom edges of the launcher. That should drag your screen over enough to see the (now huge) Play button to launch the game.
I'm having the same issue. It seems that the new launcher has more stuff than can fit in the old launcher window, and due to an oversight, they didn't increase the size of the launcher window. you can't maximize the window, you can't alt+enter the window, so there is no way to actually launch the game from the launcher as you can't actually SEE the button to do so.
EDIT: As stated above while I was writing this post, you can indeed highlight the Cryptic copyright text and drag your mouse over and down to get the rest of the stuff that's supposed to be in the window.
I'm not a big fan of game launchers, I think they are unnecessary. Several other MMOs, including WoW don't require you to use a launcher. You can launch the client directly and if there is a patch, it detects it, closes down, and runs an updater. This should be an option to negate the required use of a launcher.