An interesting question, which has been discussed for years. After a lot of research into the subject, we have come to this conclusion:
Regardless of everything else, Rasputin had a healing effect on Alexei when the best doctors available had prepared the Family for his death, at Spala. The Court had gone so far as to prepare an Imperial Bulletin, not released, to this effect. Rasputin sent a telegram, and quite miraculously, Alexei recovered. This healing effect was repeated several more times, so the Family believed it to be real and sent to them by God.
Nicholas and Alexandra had received three official reports about Rasputin, the last one from Stolypin himself. Unfortunately, they mostly contained more false information than true. People in the Imperial administration were trying to make Rasputin look even worse than he was, or were repeating the worst of the rumors without confirmation of them, in order to enrage the Tsar, but the strategy backfired when it was confirmed to Nicholas that most of the stories were just made up.
Without going into too much detail, due to space limitations here, the end result was that after the Tsar had the charges against Rasputin investigated three times, and found them to be mostly made up, he ordered that he never again be presented with these "stories" because he had become convinced that people were just making them up out of jealousy for Rasputin's closeness to the Court.
The other part of the equation is simply this: Rasputin had less actual influence over the Tsar than is generally believed. He only appeared at Court seven times, and the longest audience was 20 minutes. While he had tea with them, he never even dined with the Family, and was never once alone with The Empress.
To sum it up, Nicholas simply felt that while Rasputin had some definite character flaws, they weren't really worse than most of the men around him, who also drank and fooled around with women, Nicholas believed Rasputin was also a truly holy man of God, and that the rest of what he heard was just "sour grapes " from people jealous of his position, and Nicholas really did not feel that Rasputin had ANY impact on his decision making, so he ignored it all.