It has been proposed (or maybe just speculated) by a number of philosophers and scientists, including Richard Dawkins, that consciousness might be an epiphenomenon, but I think this cannot be the case.
According to Merriam-Webster's definition, an epiphenomenon is "a secondary mental phenomenon that is caused by and accompanies a physical phenomenon but has no causal influence itself". So, this would imply that if it is only an epiphenomenon, then consciousness can have no effect that outlasts itself.
But, if we look at what we can remember, it is surely (mostly) the memory of conscious experience. We can actually remember being conscious. It would seem impossible for us to remember being conscious if consciousness has no causal effect. Leaving a memory in the brain, however tenuous or imperfect, is certainly an effect and if it is a memory of being conscious then surely consciousness must in some way have caused it?
A counter-argument might be that the experience of consciousness arises from certain sub-conscious stimuli in the brain, and it is the sub-conscious stimuli that are recorded in memory, and can be replayed. But that still would not explain how we can recall the actual experience of being conscious.
So, we would need to take the counter argument further: we might say that having recorded the sub-conscious stimuli, a memory of the conscious experience could be reconstructed from this recording, playing back, as it were, those subconscious stimuli to generate a psuedo-memory of having been conscious. But I can't see how this could be possible, as playing back the subconscious stimuli would only re-create the experience of being conscious - we would not then recall the experience as a memory of the experience in the past, we would instead simply re-experience the whole experience as if we were going through it again.
Any attempt to define a process which can somehow create the memory of being conscious without having recorded the the actual experience itself seems impossible. Which leads to the conclusion it is the experience of consciousness which is being recorded. So, that must mean that consciousness does have some causal effect and cannot be an epiphenomenon.
2019 01 09 - Further exposition of the question, in response to @Conifold's comments
As I understand it @Conifold's - please correct me if I am wrong - position is: 1) Even if consciousness is an epiphenomenon, it would be possible to form a memory of being conscious. Therefore, being able to remember being conscious does not rule out the proposition that consciousness is an epiphenomenon.
I am going to address only the first part, the problem of forming a memory of the epiphenomenon of consciousness. In one of your comments below is an example of the memory of a reflection in a mirror: One can have a memory of the reflection in the mirror caused by that reflection. I don't think this addresses the question - my memory of an image I have seen in a mirror is not the same thing as my memory of the conscious experience of seeing an image in a mirror. It is the memory of having had the conscious experience that I am trying to pin down. Also, I am not convinced that the reflection in a mirror is a helpful parallel anyway, at least not in this context. A reflection can certainly have causal effects, not the least of which is to cause the perception by us of an illusion of space beyond the surface of the mirror. But that illusion is within the brain, it is not a property of the mirror nor the reflected light. The fact that we perceive space beyond the mirror results from the stimulus of the light (which has interacted with the physical properties of the mirror and followed the mathematical rules of reflection) generating signals from my eye that get integrated into an image in my brain.
When referring to consciousness as an epiphenomenon, my understanding is that this idea arises from our inability to detect consciousness having an influence on any decision-making in the brain or having control over physical actions of the body. At least, from my admittedly rather limited reading of the various discussions in which this point has been made, I understand this to be the key issue: can consciousness get involved in decisions to take physical action and contribute to improved survival chances, which could then explain its own evolution? If not, then we could abolish consciousness and still all go about our lives in exactly the same way, but without having any conscious experience of any of it.
I raised the original question because I think it's very likely (and, I hope, widely accepted) that memories play a part in survival. For example, the memory of being in a life-threatening situation could help me to avoid getting into that situation again, and thereby help me to survive. If memory has any purpose at all, that seems like a good one. Therefore, I surmised, if we can show that consciousness does have an effect on memory, then it is quite possible that consciousness could help survival, albeit indirectly. Clearly there would be many other implications, but those are beyond the scope of the question.
That's why I have focused on the question of whether can we tell if consciousness has an effect on memory. My proposition is: if it is possible to remember the experience of being conscious, then the only possible source of the memory is the conscious experience itself. It is impossible for any other function of the brain to generate a memory of the experience of being conscious, because no other function of the brain has access to conscious experience to be able to create a memory of what it feels like. If you say there could be another function of the brain that "watches" conscious experience in some way, and records what it "sees" so that the experience can be played back, then my response would be that you are simply interposing a layer of the mechanism of memory (which must exist in some form anyway), and any such mechanism does not detract from the point that it is the conscious experience which has caused the memory. That memory is a change in the state of the brain, and it may have an effect, however slight, on subsequent actions. That tenuous connection, however slight, torpedoes the epiphenomenon idea.
