Yes a possible connection between entropy and measurement has been explored for a long time. Overall, the results remain inconclusive because measurement in quantum mechanics remains a subtle issue and it is not fully agreed what is the best way to interpret what goes on.
First I should correct slightly your statement about entropy. The second law statement is not that the entropy of any system can never go down, it is that the entropy of an isolated system cannot go down. So when a system is not isolated, for example because it interacts with another system such as a microscope, then in principle its entropy could go up or down, depending on whether there is any exchange of entropy between the two systems.
Having said that, the kinds of physical process that are involved in recording measurement outcomes are the kinds that are indeed irreversible from a thermodynamic point of view. That is, they involve a net increase in the entropy of an isolated system. An example could be the sequence of sparks or current impulses generated in a particle detector, or the chemical process whereby spots are registered on photographic film, or the chemistry in the retina of an eye, and things like that. So it is reasonable to say that there is a connection between thermodynamic irreversibility and measurement of quantum states.
But the difficulty is that quantum theory itself can in principle be applied to describe the kinds of complex avalanche-like processes we are discussing, and quantum theory proposes that everything follows Schrodinger's equation, which amounts to saying that everything is reversible in principle. So this is, roughly, where the subject lies. Some people think Schrodinger's equation 'rules' and there is no real irreversibility, just a technological difficulty in controlling complex processes. Other people think Schrodinger's equation is not ruling in that way, but rather showing how the various parts of an overall irreversible process combine, leading to the probabilities for the available outcomes.