We are told in the Bible that the serpent deceived Adam and Eve and that same serpent was used by Moses as sign of healing to those were bitten by snakes in the wilderness. What is the significance of the serpent in the bible in connection to the faith of a believer?
Paul teaches that :
Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned: [Romans 5:12 KJV.]
And he also teaches that :
For he hath made him to be sin for us, who knew no sin. [II Corinthians 5:21 KJV.]
Jesus said :
And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up. [John 3:14 KJV.]
Sin entered the world and when sin is judged, the judgment focuses, first, on the origin of sin. A spirit other than God was the Tempter who was, originally, responsible. Then, also, the woman was deceived by the spirit who behaved in a serpentine manner. Then, further, the man transgressed, influenced by the woman.
In the wilderness, the plague (of death) was only stayed when the serpent of brass (a figure of judgment) was raised up by Moses on a 'pole', Numbers 21:9, or as an ensign (see the Hebrew meaning of 'pole' Strong 5251).
Likewise was the Son of man lifted up, whereupon he was made sin (here, the Greek word for 'made' covers both 'do' and 'make' and can be also translated 'effected' or 'caused', see Strong 4160).
In judgment, Jesus Christ, the Son of God - in manhood - was effectively caused to be sin - sin was contained within him. And with sin contained within him, he died. Thus was sin eradicated : in him, through death.
As Paul teaches :
Our old man is crucified with Him, that the body of sin might be destroyed. [Romans 6:6 KJV].
This is received by faith, as the question states - 'the faith of the believer'.
In the same passage in which he mentions the serpent in the wilderness, Jesus makes this clear, that faith is the way of salvation :
Whosoever believeth in him should not perish, but have everlasting life. [John 3:16 KJV.]
Jesus Christ - he without whom nothing was made that was made (John 1:3) - is he who took responsibility for the entering of sin into the world, receiving effectively in himself - in a clean humanity - the spiritual consequences of sin's entry into the world, in order that the world, through him, might be saved.
For God sent not his Son into the world to condemn the world; but that the world through him might be saved. [John 3:17 KJV.]