Having recently upgraded my laptop I also ran into the same problem, and as this screenshot highlights, by default GATE Developer isn't at all usable on a HDPI screen.
A quick hunt around the web and you'll find all sorts of suggestions for getting Java 8 to work nicely with HDPI screens, but try as I might I couldn't get any of them to work for me; I'm running OpenJDK 8 under Ubuntu 16.04. Fortunately HDPI support is going to be built into Java 9. Unfortunately Java 9 still hasn't been officially released so you need to rely on an early access version.
In theory it should have been easy for me to see if Java 9 was a solution, but unfortunately the version of Java 9 in the Ubuntu 16.04 repositorie causes a segfault as soon as you try to run any Java GUI program making life more difficult than it needs to be.
The solution is to install the Oracle early access build of Java 9. You can either download the JDK manually, or follow these instructions under Ubuntu to install from the very useful Web Upd8 repository. Either way once installed, launching GATE gives a usable UI.
Unfortunately this isn't quite enough to solve the problem. Under the hood Java 9 introduces a modular component system (often referred to as Project Jigsaw) which includes new rules on encapsulation. The issue is that one of the libraries GATE uses for reading and writing applications, XStream, uses a number of tricks to access internal data that are prohibited under the new rules. The result is that you can't load or save applications which makes the GUI kind of pointless. Fortunately there is a command line option you can pass to the JVM that allows you to bypass the encapsulation rules. So to get GATE to work properly with Java 9 you need to add
--permit-illegal-accessto the command line. When launching the GUI this is easy to do by adding the flag as a new line in the gate.l4j.ini file which you will find in the GATE home folder.
There are two important things to note. Firstly this fix is only temporary as the command line flag will be removed in a later version of Java, and secondly depending how you are deploying GATE it can be difficult to alter the command line arguments (for example if deploying as a web app). Once Java 9 is officially released we'll look again at this problem to find a more permanent solution. Until then this gives you a way of using GATE on a HDPI monitor, but where possible (i.e. only on a HDPI monitor when you need the UI) we'd still advise using Java 8 and this hack as a last resort.