Learning new things after coursework duration of PhD In STEM (In USA), the coursework duration of the PhD typically lasts about 1-2 years beyond that, students take 0 (or in certain cases, very few) courses in their subsequent PhD years. I work in applied math with a lot of applications in every field imaginable : Finance, Economics, Energy, Healthcare, Services etc. 

During my coursework phase (owing to the breadth requirement), I was made to take courses in different departments and each time I attended a course in a new department I would think "Wow, this field could really benefit from extensions of my research". I would chalk out a few ideas and read up a few papers, find out that it's already done but be glad that I generated a new idea nonetheless. 

Now, I am nearing the end of my coursework phase and I am wondering : How do I keep my ability to "see" applications in fields other than mine alive?

Global Question : After a student's coursework phase, how does he stay aware of applications of his research field besides the ones discussed in papers?

My advisor encourages me to attend as many seminar talks as possible (in all allied fields) which I do but it's not the same level of enjoyment.
First of all, unless there is a policy against it, you should feel free to enroll in subsequent courses even though you are done with your requirements.  The caveat is that this will take time, and your primary responsibility after finishing those requirements is to do research in your specific area to make progress on your PhD.

To a first approximation, a PhD program is designed to have you focus on one particular problem until you are the only person in the world that completely understands that problem.  That is not to say that you won't find research inspiration from those other seminars, and in fact you might find that you will tailor your research based on one (probably not more) of those other fields.  But, if you continue an academic career after you get your PhD, you'll have more flexibility to collaborate outside your particular field, and that is probably a better time to do it than right now.

To answer your specific question: you're on the right track by going to seminars, and if you can justify the time, taking subsequent classes is not out of the ordinary.  You could also get on mailing lists for other fields that interest you, and nothing is stopping you from becoming a member of professional organizations outside your immediate field (and, they're cheap when you're a student!).  You could also consider collaborating with folks from other departments on projects, bearing in mind that you'll need to be able to differentiate your contributions if you want to use that work in your dissertation.