Why are we to doubt this passage? Don't we know for a fact, from the testimony of scripture itself, that the apostolic generation did have these experiences? Don't we know they cast out demons over and over and over? That they spoke in tongues? That Paul picked up a poisonous viper and shook it off without harm? Don't we know they healed the sick miraculously over and over and over? Why is there any doubt about this passage? Why do the scholars choose to believe the corrupted texts preferred by Westcott and Hort? Answer: They are trusting in their fallible minds instead of in GOD.
9 Now when he was risen early on the first day of the week, he appeared first to Mary Magdalene, from whom he had cast out seven demons. 10 She went and told them that had been with him, as they mourned and wept. 11 And they, when they heard that he was alive, and had been seen of her, disbelieved. 12 And after these things he was manifested in another form unto two of them, as they walked, on their way into the country. 13 And they went away and told it unto the rest: neither believed they them.
14 And afterward he was manifested unto the eleven themselves as they sat at meat; and he upbraided them with their unbelief and hardness of heart, because they believed not them that had seen him after he was risen. 15 And he said unto them, Go ye into all the world, and preach the gospel to the whole creation. 16 He that believeth and is baptized shall be saved; but he that disbelieveth shall be condemned. 17 And these signs shall accompany them that believe: in my name shall they cast out demons; they shall speak with new tongues; 18 they shall take up serpents, and if they drink any deadly thing, it shall in no wise hurt them; they shall lay hands on the sick, and they shall recover.
19 So then the Lord Jesus, after he had spoken unto them, was received up into heaven, and sat down at the right hand of God. 20 And they went forth, and preached everywhere, the Lord working with them, and confirming the word by the signs that followed. Amen
Yes, I know the problem with this text comes in with applying it to the church after the apostolic age, but taking it just as written there is no need to insist on that application, since we know it is almost word for word prophetic of the ministry of the apostles. We know mostly of Paul plus some accounts of a few others, but from their experience we shouldn't have any trouble surmising that all the apostles experienced such miraculous powers. Some other time I might even be willing to argue that they are still available today under certain circumstances, but for now just on the face of it this much-disputed passage in Mark ought to be regarded as unimpeachably the word of the Lord.
And then if you read the evidence for its legitimate transmission in the following you should have no more doubts at all. This is a page about Mark 16 from Dr. Thomas Holland's Crowned With Glory which gives historical reasons to accept the passage and gets at the mutilating effects of the fleshly mind on God's word:
Most scholars believe the original ending to Mark's Gospel has been lost.  If this is true, the concept of preserving the words of Scripture is forever annihilated. The words cannot be preserved and lost at the same time. However, textual scholars usually call for its inclusion even if they question its originality. Dr. Bruce Metzger departs from the maxim of modern textual critics, Brevior lectio potior (the shorter reading is preferable), and supports the longer ending even though admittedly he does not regard the passage as genuine. He considers it to be a legitimate part of the New Testament because of its traditional significance to the body of Christendom.  The passage is not contained in the Alexandrian texts, minuscule 2386, the Syrian Sinaitic Version, and a few other translations.
However, it is in many of the Greek uncials (A, C, D, K, X, D, Q, and P) dating between the fifth and ninth centuries. It is also contained in later dated Greek minuscules (137, 138, 1110, 1210, 1215, 1216, 1217, 1221, and 1582). It is the reading found in the majority of Old Latin texts as well as the Coptic versions and other early translations. Finally, it is cited (at least in part) by many of the early church fathers such as Justin (165 AD), Tertullian (220 AD), Hippolytus (235 AD), Ambrose (397 AD) and Augustine (430 AD). 
In 177 AD Irenaeus wrote Against Heresies. In it he cites from Mark 16:19, establishing that the longer reading was in existence at this time and was considered canonical, at least by Irenaeus:
Also, towards the conclusion of his Gospel, Mark says: "So then, after the Lord Jesus had spoken to them, He was received up into heaven, and sitteth on the right hand of God; " confirming what had been spoken by the prophet: "The LORD said to my Lord, Sit Thou on My right hand, until I make Thy foes Thy footstool." Thus God and the Father are truly one and the same; He who was announced by the prophets, and handed down by the true Gospel; whom we Christians worship and love with the whole heart, as the Maker of heaven and earth, and of all things therein. (3:10:5).
The difference here is extremely important. If we conclude that this passage is not authentic, then we must question what happened to the original ending of Mark. It is not logical that the Gospel would end at this place so abruptly. Nor is it likely, as some scholars have suggested, that the Gospel was never finished, calling biblical inspiration into question. The conclusion held by most textual scholars, whether liberal or conservative, that the original ending has been lost over the passage of time certainly denies the doctrine of biblical preservation.
Seems to me the W&H defenders need to shake off their trust in the scholarly establishment, or more to the point shake off their fleshly intellect and open their spiritual eyes! If you exercise your fleshly mind on these things without immersing yourself in prayer in the fear of God and abiding in Him, you will end up discarding His very word and missing the spiritual riches He has given us.
Doubt about the passage is based completely on accepting Westcott and Hort's text and their despising of the Textus Receptus which was the traditional text on which the KJV was based. It also suggests the liberal prejudice against the supernatural we're so familiar with today. Both inclinations have brought about the mutilation of the Biblical text which had been passed down through the centuries, a mutilation apparently accepted today even by the most conservative scholars, even those who are usually alert to this kind of destruction. Following Westcott and Hort's mugging of the Bible, today conscientious studious pastors of even the most Bible-focused churches determine the canon of Holy Scripture by the flesh rather than the spirit. That is why they have either done away with Mark 16:9-20 or hold it in oh-so-fussy head-proud "doubt."
I know so many sincere Christians who will defend the Westcott and Hort profanation of God's word. Otherwise good preachers trust in their scholarly training and their carnal intellectual strengths for judging God' word. The church is so weak compared to what it seems to me the Bible shows us is possible and desired by the Lord. The flesh can preach the Bible, can even preach spiritual truth (and I am susceptible to the same mistake), but only what is preached in the spirit has value and has power to save.