What should I do if a fluid clips through a cup while it sits in it?

I’ve had this problem for a while now.

Is increasing the thickness of the cup and the domain resolution the only thing I can do?

Increasing the thickness is the first and easiest thing to do. And if you’re worried about the ugliness of your cup with super-thick walls, just don’t forget that you’re not obliged to render it… In other words, stop trying to adapt your final object to the fluid sim and just build simplified obstacles (set not to render). You will gain a lot of time in the baking and, in the final render, your cup holding the liquid can be as thin as paper if you want.