I've seen $k\in \emptyset$ written in the context of empty sum. I read, $k\in \emptyset$ as "$k$ is an element of the set which has no elements." To me, this sounds like a contradiction. If the empty set contains the element $k$ it is no longer empty.
Also, how do you arrive at the conclusion that because $k\in \emptyset$, $k$ does not exist? To say that $k$ is in the $\emptyset$ is a false statement, therefore nothing follows from it. If we want to say that $k$ does not exist we simply write $\neg \exists k$.
Since $k\in \emptyset$ is widely used there must be an explanation. Can you clarify how you rationalize the contradiction that empty set is not really empty? Does the set theory has its own logic where contradiction is legal?
Edit
I'm sorry it was not clear from the question that $k$ is a summation index and as such it cannot be a free variable. $k$ takes natural numbers as its values $k\in \mathbb{N}$.
I think, therefore, answers claiming that $k\in \emptyset$ cannot be false because $k$ is a free variable are not acceptable. Please correct if I misunderstand.