I'm currently investigating this issue on Windows. Here's what I found:
Most Swing Look & Feels don't support high DPI at all, not even Nimbus even though it's supposed to be scalable. I found some old blog posts saying that Nimbus might eventually offer high DPI scaling, but apparently that never happened.
The one exception is System LAF but its default font is ~10% smaller than the actual system font size, at all DPI settings. Moreover, System must be selected explicitly as described here: http://docs.oracle.com/javase/tutorial/uiswing/lookandfeel/plaf.html
There's no single scaling factor that you could set in Swing, either. The specific LAF has to provide code to handle scaling. So the best you can do is select System and hope it's good enough.
However, JavaFX does correctly and automatically scale all the way up to 150% on my system. If at all possible, I suggest you use JavaFX to build your GUI.
edit: I made a couple small test programs and took comparison screenshots for various GUI frameworks, Swing themes, and DPI settings. This might be informative for people reading this question: http://kynosarges.org/GuiDpiScaling.html