What is the basis for the doctrine of the Trinity in the Book of Mormon?

There are a great many verses in the Book of Mormon that on the surface appear to support the trinity, even more than there are in the Bible. For example:

Alma 11:44 ... shall be brought and be arraigned before the bar of Christ the Son, and God the Father, and the Holy Spirit, which is one Eternal God, to be judged ...

3 Nephi 11:25 Having authority given me of Jesus Christ, I baptize you in the name of the Father, and of the Son, and of the Holy Ghost. Amen.

3 Nephi 11:27 And after this manner shall ye baptize in my name; for behold, verily I say unto you, that the Father, and the Son, and the Holy Ghost are one; and I am in the Father, and the Father in me, and the Father and I are one.

There are a few more in the Pearl of Great Price and the Doctrines and Covenants, but seeing as how you did not ask about those, I will not quote those at this time. However, it should be realized, after reading through the Book of Mormon and doing a thorough study of the the various concepts of the godhead, that what the Book of Mormon ACTUALLY teaches is modalism. For example:

Alma 11:38-40

38 Now Zeezrom saith again unto him: Is the Son of God the very Eternal Father?

39 And Amulek said unto him: Yea, he is the very Eternal Father of heaven and of earth, and all things which in them are; he is the beginning and the end, the first and the last;

40 And he shall come into the world to redeem his people; and he shall take upon him the transgressions of those who believe on his name; and these are they that shall have eternal life, and salvation cometh to none else.

Ether 3:14 Behold, I am he who was prepared from the foundation of the world to redeem my people. Behold, I am Jesus Christ. I am the Father and the Son...

Ether 4:7 ... saith Jesus Christ, the Son of God, the Father of the heavens and of the earth, and all things that in them are.

Ether 4:12 ... he that will not believe my words will not believe me—that I am; and he that will not believe me will not believe the Father who sent me. For behold, I am the Father, I am the light, and the life, and the truth of the world.

etc, etc.............

So, as you can see, there are several verses in the Book of Mormon to support the theory of the trinity and many more modalistic verses, which, if one is not familiar with the concepts of the godhead, one could misinterpret as support for the trinity.

For more notes, check out the notes I took when I went through the book of mormon, the pearl of great price and the doctrines and covenants http://dukeofmarshall.blogspot.com/p/mormon-notes.html

The question has a false premise. LDS hold that the Holy Bible is also scripture. Therefore it is not necessary for the Book of Mormon to teach that God is Trinity: it is only necessary that the Bible teach it, and the Book of Mormon not contradict it. Very many people consider that the Bible does teach that God is Trinity.

With a fair bit of searching, I could find only one source associating the Community of Christ's belief in the trinity with verses from the Book of Mormon. The verse is 3 Nephi 11:36:

And thus will the Father bear record of me, and the Holy Ghost will bear record unto him of the Father and me; for the Father, and I, and the Holy Ghost are one.

In order to join the National Council of Churches (NCC), The Community of Christ had to make several concessions, including reducing it's reliance on the Book of Mormon. A 2010 NCC report made the following statement about the Community of Christ's belief in the Book of Mormon:

Leaders of the COC... affirmed without qualification that the Bible is the foundational, authoritative Scripture of the church. They acknowledged that “the Book of Mormon is in our DNA,” and suggested that it confirms God’s revelation in Jesus Christ as testified in the Bible. But it is not, in any sense, equivalent to the Bible in the life of their communion.
Subscription to its teaching is not required for membership or ordination. While the Book of Mormon is sometimes used in worship, there are parts of the COC that seldom refer to it.

Considering that the Community of Christ has made the Book of Mormon optional, I doubt they feel the need to find support for any of their doctrine in it beyond general support for Christ.