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.