What is the basis of the LDS (Mormon) belief that the Apostle John never died?

The basis for this teaching is, as noted in another answer, is found in John 21:21-23, where Jesus, speaking to Peter, sidesteps a question regarding John's future death. More direct teaching is found in Doctrine and Covenants 7, where Jesus is quoted as saying:

¹ And the Lord said unto me: John, my beloved, what desirest thou? For if you shall ask what you will, it shall be granted unto you. ² And I said unto him: Lord, give unto me power over death, that I may live and bring souls unto thee. ³ And the Lord said unto me: Verily, verily, I say unto thee, because thou desirest this thou shalt tarry until I come in my glory, and shalt prophesy before nations, kindreds, tongues and people.

The LDS Church's summary of this is as follows:

The latter-day scriptures clarify that John did not die but was allowed to remain on the earth as a ministering servant until the time of the Lord’s second coming. (Guide to the Scriptures)

So yes, John is on the earth. As to his location, however, we don't have much in the way of details. The student manual to the Doctrine and Covenants, however, reports two appearances. The first is that Joseph Smith reported that the Apostle John (along with Peter and James) appeared to him and Oliver Cowdery in 1829. LDS historians generally hold that John, unlike the others, was not resurrected for this appearance, but rather "translated," that is, physically changed by God from a mortal to immortal being. As McConkie writes:

At that appearance, Peter and James were resurrected beings; John was translated. (Mormon Doctrine, pg. 572)

A separate appearance was reported by Heber C. Kimball at the dedication of the Kirtland Temple in 1836. (Whitney, Life of Heber C. Kimball, pp. 104)

More generally, the manual continues, there is one teaching from Joseph Smith that provides insight into why John's location is such a mystery:

In a conference of the Church on 3 June 1831, the Prophet Joseph Smith taught concerning John’s ministry: “John the Revelator was then among the Ten Tribes of Israel who had been led away by Shalmaneser, king of Assyria, to prepare them for their return from their long dispersion” (History of the Church, 1:176).

This is a reference to the Ten Lost Tribes of Israel, which were dispersed following the Assyrian conquest, and whose location is unknown. There is "speculation" that they went to the north, but "The Lord has not seen fit to reveal their location, [...] and until he does so, it is useless to try to identify their present locality." (Old Testament Student Manual, D5)

Thus, from Smith's teaching, John is (or at least was) "among" this dispersion of Israel, but we aren't meant to know where that dispersion is.

As mentioned in previous answers, the appointment of John to remain on the earth to preach the Gospel is analogous to the three Nephite apostles Jesus chose to do the same while visiting America. In fact, He said to them:

...ye have desired the thing which John, my beloved, who was with me in my ministry, before that I was lifted up by the Jews, desired of me.

3 Nephi 28:6

While records concerning the whereabouts of John are scarce, records concerning the Three Nephites are plentiful, such as as the following:

25 Behold, I was about to write the names of those who were never to taste of death, but the Lord forbade; therefore I write them not, for they are hid from the world.

26 But behold, I have seen them, and they have ministered unto me.

27 And behold they will be among the Gentiles, and the Gentiles shall know them not.

28 They will also be among the Jews, and the Jews shall know them not.

3 Nephi 28:25–28

If one can accept that the Apostle John and the three unnamed Nephite apostles received the same calling, rights, protection, and privileges, then it can be assumed that similarly John is hid from the world. John will be among the Gentiles and they will know him not, and John will be among the Jews and they will know him not.

Consider studying 3 Nephi 28 for more information and prophecies about the Three Nephites.

Doctrine and Covenants Section 7

Revelation given to Joseph Smith the Prophet and Oliver Cowdery, at Harmony, Pennsylvania, April 1829, when they inquired through the Urim and Thummim as to whether John, the beloved disciple, tarried in the flesh or had died. The revelation is a translated version of the record made on parchment by John and hidden up by himself.

1 And the Lord said unto me: John, my beloved, what desirest thou? For if you shall ask what you will, it shall be granted unto you.

2 And I said unto him: Lord, give unto me power over death, that I may live and bring souls unto thee.

3 And the Lord said unto me: Verily, verily, I say unto thee, because thou desirest this thou shalt tarry until I come in my glory, and shalt prophesy before nations, kindreds, tongues and people.

4 And for this cause the Lord said unto Peter: If I will that he tarry till I come, what is that to thee? For he desired of me that he might bring souls unto me, but thou desiredst that thou mightest speedily come unto me in my kingdom.

5 I say unto thee, Peter, this was a good desire; but my beloved has desired that he might do more, or a greater work yet among men than what he has before done.

6 Yea, he has undertaken a greater work; therefore I will make him as flaming fire and a ministering angel; he shall minister for those who shall be heirs of salvation who dwell on the earth.

7 And I will make thee to minister for him and for thy brother James; and unto you three I will give this power and the keys of this ministry until I come.

8 Verily I say unto you, ye shall both have according to your desires, for ye both joy in that which ye have desired.