How did the people know that John the Baptist was a prophet? What reasons gave the people to believe that John The Baptist actually was a prophet and wasn't a hoax? Was it simply the Holy Spirit calling on souls or was there some other kind of proof?
I wouldn't overthink this one, LukeoX. 

God obviously called John to be a prophet and a forerunner to His Son Jesus, the Christ (Luke 1:17). Consequently, in obedience to God's call John presented himself to the people as a witness, called by God to be a testimony to the Light. Put differently, John was a light to the people, and Jesus was the Light of the world (see John 1:4-9; cf. 8:12). John's little light paved the way for the big Light. 

When people believed John's message, they proved the sincerity of their repentance by submitting to his baptism. They then became John's disciples (see Matthew 9:14; 11:2; Mark 2:18; Luke 5:33; 7:18; John 1:35; 3:25; and 4:1). The angel Gabriel told John's father that his yet-to-be-born son would come on the scene "in the spirit and power of Elijah" (Luke 1:17). In other words, John's preaching ministry would bear fruit, just as Elijah's did. 

Now, what all was involved in being one of John's disciples is not exactly clear to me, nor does the Bible say too much about that. Unlike Jesus, however, I don't think John was called of God to form a denomination such as--oh, I don't know--Baptists perhaps!?
As the Christ's forerunner, he sort of faded into the background shortly after Jesus came on the scene. After all, John admitted humbly, 


  "He must increase, but I must decrease" (John 3:30).


Not too terribly long after John made that statement, he was beheaded by King Herod. Interestingly, however, just before he died John told his disciples to ask Jesus, 


  "'Are you the Expected One, or do we look for someone else?'" (Luke 7:19). 


Despite John's greatness in Jesus' eyes ("among those born of women," Jesus said, "there is no one greater than John"), even he struggled with doubts about the identity of his cousin(?), the son of his mom's relative, Mary, whom John's mom called "the mother of my Lord"! (Luke 1:43).