Has any Christian Apologist ever published arguments for the existence of Satan, demons and their ongoing influence over human matters?

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Yes

CS Lewis addresses it in many places, though perhaps most thoroughly in Mere Christianity. An abbreviated discussion of Lewis' thoughts can be found here.

Simply searching "proof of the existence of satan" will get you any number of articles, treatises, blogs, and more on the subject.

Others, as mentioned in the question and its comments, do not bother to prove the existence of satan through logic. Consider the following quote from Jesus the Christ, as written by James E Talmage (of the Latter-Day Saints):

Many modern writers have attempted to explain the phenomenon of demoniacal possession; and beside these there are not a few who deny the possibility of actual domination of the victim by spirit personages. Yet the scriptures are explicit in showing the contrary... In this matter as in others the simplest explanation is the pertinent truth; theory raised on other than scriptural foundation is unstable. Christ unequivocally associated demons with Satan...

Apologists often address things using a domino theory. "If A is true, then B must be true." If the Bible is the word of God, then satan and demons must exist, for they are explicitly discussed in the Bible. The apologist thus works to convince the reader that the Bible is the word of God, rather than attempting to address every teaching it contains.

This leads to a conundrum: what if the reader can accept the apologist's arguments in some ways, but struggles to reconcile something not addressed by the apologist? In this case, let's say the apologist makes good arguments for the Bible being the word of God, but does not address satan or demons, which the reader questions the validity of. This makes the reader doubt the apologist's other arguments, not because they were not valid, but because, in the reader's eyes, they were not complete.

The apologist would say this is illogical. Either the Bible is the word of God, or it isn't, and if a preponderance of evidence convinces someone it is, then anything left unaddressed can be accepted on the basis of God being involved in the Bible throughout.

The reader might argue that the apologist needs to prove the entirety, for if the Bible is mostly correct, but got it wrong in regards to satan and devils, then it can't be the word of God as described within its own pages.

To use an analogy, if someone can explain a car's combustion engine sufficient that they convince a reader it works, they need not explain the suspension to prove that the thing a person sees driving down a road is a car and that it has a suspension system. This will not satisfy every reader, and is perhaps a failing that more apologists should address, but as mentioned above, there are some addressing it.