I guess it is easy to take a quote from 100 years ago, but more difficult when you put it in context of the time. The “Spiritual movement “ was at its peak during Tesla’s hey day. It was a big topic and included in scientific discussions and journals of the day. It wasn’t until people like Houdini (the original James Randi) began debunking the claims that the spiritual movement declined.
So in the heightened religious/spiritualist moment, the quote was relevant.
Of course it is silly by our standards, but hey, the best surgeons in the world did not know enough to wash their hands before surgery, and that was only 100-150 years ago.

I thought Tesla was supposed to be smart? If, indeed, this is an authentic quote(which I haven't been able to verify), it's idiotic.

Kirlian Photography produces interesting and colourful images. I think that anything added to those images by way of interpretation is very subjective and is essentially no more different or objective than a group of religious people espousing their belief in a supreme being.

How would non-physical phenomenon be measured? What sort of nonphysical phenomenon is he referring to?

Social sciences study nonphysical phenomenon already. The create situations and observe subjects reaction. They may also take surveys from subjects to see what they report about their experience while participating in the research. This sort of research has been going on for many decades now. Somehow, I don't believe this is the kind of nonphysical phenomenon the quote had in mind, so answering the original question become important.

Putting aside the spiritual vs scientific labels, let's just do good science without a prejudiced mindset and see what we come up with. What's so wrong with that, sounds like what science is based upon?

This remains to be seen.

What a load of crap!

Yes and No, imo, BUT when we start to " Think outside of the box/boxes then and only then shall we truly learn to understand the Universe, etc, around us." - William Anthony, 2017

Gravity, electricity, electrostatic and electromagnetic fields, magnetism, radiation. These were all mystical and nonphysical phenomena until we investigated them and found ways of measuring and reproducing them. And the description of the arc of the covenant which supposedly held some spirit sounds suspiciously like a leyden jar (google it), a box plated inside and out with gold foil.

I do not agree. People seriously involved in rigorous scientific research are interested in the discovery of the truth based on evidence. If there was any use for the scientific process in researching non-physical phenomena, they would have done it long time ago without the prompting of Mr. Tesla. Serious scientists follow their research wherever they lead them to. Tesla, like Edison and others, were not scientist. They were very intelligent and curious practical inventors with very practical and utilitarian results, not a small feat, but not necessarily scientists dedicated to pure research with strict methodology and peer reviewed publications.
In making his assertion, Tesla begs the question of why is it that he did not dedicate more time to those phenomena and made the huge scientific advances himself? In fact, he dedicated himself to the total oposite; practical material physical phenomena resulting in practical utilitarian inventions.

I think science will find there is nothing to study with the non-physical.