How did ancient mystery religions impact Christianity?

Yes. All religions and cultures build and borrow from what went before. There are numerous contemporary examples, Christmas in western culture for example. Helios, the Greek sun god no doubt inspired many derived religions including Mithraism which could be considered an aberrant form of Christianity.

Mormonism is a fairly recent example demonstrating this principle of building on rather than replacing.

I read the book referred to here and found it interesting, I believe it's directly related to your question:

http://en.wikipedia.org/wiki/The_Jesus_Mysteries

Evidently, early Christians (2nd Century) were well aware that Christianity was strikingly similar to prior Mystery religions, and responded:

Essentially, these early Christian apologists were saying, "So what, our religion looks similar to your religions, but we did not copy from you. The Devil mimicked our religion in your religion in anticipation of the advent of our religion". Incredible as this may seem, this was nonetheless the standard excuse given by the early church apologists; one which has basically not changed at all in the entire two thousand years of the evolution of Christianity.

I don't know much about these mystery cults, so I can't provide specific examples of what was supposedly contributed or argue for or against specific claims, but after reading over that wordy paragraph a few times I can at least explain what it's saying in the historical context of Christianity:

Through the 1st to 4th century, Christianity stood in direct competition for adherents with the mystery cults, insofar as the "mystery cults too [were] an intrinsic element of the non-Jewish horizon of the reception of the Christian message".

When Christianity came onto the scene, it was a spin-off of Judaism in a backwater Roman province. One of the earliest controversies in the church was whether Christ's message was intended for Jews only or for Gentiles (non-Jews) also. The latter view prevailed, and so Christianity became a very evangelistic movement as it expanded beyond Judea.

In reaching out to followers of other religious cultures, Christianity necessarily interacted with the beliefs and practices that were already out there, in some cases using what people already knew to teach them about Jesus. As a small example, in the book of Acts the apostle Paul used the Athenian altar to an unknown god to proclaim the revelation of Christ to the world.

They too were "embraced by the process of the inculturation of Christianity in its initial phase", and they made "their own contribution to this process".

No cultural elements develop in a vacuum. Christianity took a while to establish itself in the Roman culture and beyond as a tradition in its own right- and in doing so it borrowed some things from its interaction with the mystery cults that were already present in contemporary society. In a way, some of these elements piggybacked on the growing Christian movement to become a part of the culture.

In Klauck and McNeil's opinion, "the Christian doctrine of the sacraments, in the form in which we know it, would not have arisen without this interaction; and Christology too understood how to 'take up' the mythical inheritance, purifying it and elevating it".

What our friends are asserting here is that the sacraments, especially important in the Catholic tradition, are relics of these mystery cults which were imported into early Christian tradition and "hallowed" in a sense by relating them to the theology of Jesus. The use of the sacraments, then, are themselves an example of how early Christianity was influenced by the mystery cults.

First of all, these subject is a lot easier to research if you know that the “mystery religions” being referred to here are the various incantations of Gnosticism.

Gnosticism was was not unique to that erra of history. Versions are documented long before the time of Christ and it can still be found in various flavors today. However it did perhaps reach its popular climax about the same time Christianity was spreading through the 1st century modern world.

The various ideas of Gnostic thought had a profound formative effect on early Christianity. However this effect was more reactive than assimilitive. The teachings of Christianity stand in stark contrast to all forms of Mysticism. Yet, as is ever the case, there are plenty of folks willing to promote various brands of syncretism. During the first couple of centuries, Gnostism and the various Gnostic cults that sprung up were the first major heresies Christianity had to deal with.

You will in fact find that much of the teaching of the early Christian apostles was specifically directed against Gnostisms. For example, the shorter letters written by the Apostle John were directed squarely against a prevalent Gnostic teaching that Jesus did not actually have a physical body but was just a vision, something like a hologram projected into this physical world. Contrary to modern perceptions, the first obstacle that Christianity had to overcome was not convincing people that Jesus was God but convincing people that God actually took on flesh and became a man.

The mystic religions of the time said that this was shameful, that a pure holy God would never defile himself by becoming fleshly. John explained the Christian doctrine to the contrary starting his letter with these words:

1 John 1:1-2 (ESV)
1  That which was from the beginning, which we have heard, which we have seen with our eyes, which we looked upon and have touched with our hands, concerning the word of life— ²  the life was made manifest, and we have seen it, and testify to it and proclaim to you the eternal life, which was with the Father and was made manifest to us—

In other words, Jesus was touchable. His body was real. He goes on to treat ideas such as “light” that were often used in mystical ways, showing that in Christianity their real meanings were quite concrete.

Early Christian history is littered with evidence of their opposition to mystic teachings. According to some popular mystic ideas, salvation was to be freed from this physical world: united with a spiritual dimension by discarding the shells that we are trapped in. Christianity stood in opposition to this, claiming salvation through the work of God redeeming both the spiritual and eventually the physical worlds.

If you follow the development of Christian creeds, you will find the points that the church felt they needed to emphasize and expand on in each successive expression of their core beliefs were usually reactions to some popular mystic idea having found its way into the practice of Christians of the time. You can find some examples of this progression in my answer to a question about the creeds here.

Martin Luther King Jr wrote a paper about this topic in 1950 titled The Influence of the Mystery Religions on Christianity. His conclusion is that you couldn't deny some influence on Christianity but that it likely wasn't intentional copying of rites and traditions.

There can hardly be any gainsaying of the fact that Christianity was greatly influenced by the Mystery religions, both from a ritual and a doctrinal angle. This does not mean that there was a deliberate copying on the part of Christianity. On the contrary it was generally a natural and unconscious process rather than a deliberate plan of action. Christianity was subject to the same influences from the environment as were the other cults, and it sometimes produced the same reaction.

King also proposes that the mystery religions had an even greater influence by preparing the people to understand Christianity.

The greatest influence of the mystery religions on Christianity lies in a different direction from that of doctrine and ritual. It lies in the fact that the mystery religions paved the way for the presentation of Christianity to the world of that time. They prepared the people mentally and emotionally to understand the type of religion which Christianity represented.

As mentioned in another answer, authors of The Jesus Mysteries claim Christianity was a product of the mystery religions (they aren't the only ones to have made this claim). But that proposal is irreconcilable with the Jewish origins of Christianity and prophecies of a Messiah, as critics of the book pointed out.

ReligionFacts.com proposes the idea that the similarities between mystery religions and Christianity may best be explained by them developing in roughly the same time of history. It's an interesting hypothesis but it seems flawed to me because mystery religions trace their origins to over 1000 years before the time of Christ (perhaps as much as 1600 years before Christ). Though mystery religions is a broad term so it's possible they were referring to more recent religions.

The only people I could find that denied any connection between the mystery religions and Christianity were Christian pastors.

In conclusion, I think that King's analysis is the most reasonable. It acknowledges both the Jewish origin of Christianity as well as the fact that mystery religions and Christianity existed during the same time period.