This passage can be understood two ways. One way is to assume that Jesus followers are expected to handle deadly snakes to prove to unbelievers that they are Christians. Another way to understand this passage is to be reassured that when Christians accidentally come in contact with poisonous snakes, God will miraculously protect them.
In looking at the subsequent history of the New Testament Church, the Bible records only the latter explanation—that if a true Christian accidentally picks up a poisonous snake and is even bitten by that snake, God can intervene to miraculously spare that persons life.
Such an experience happened to the apostle Paul. After being shipwrecked and escaping to the island of Malta, Paul was bitten by a deadly snake. The natives thought he would die, but God miraculously intervened and Paul was not harmed (Acts 28:1-6 Acts 28:1-6  And when they were escaped, then they knew that the island was called Melita.
 And the barbarous people showed us no little kindness: for they kindled a fire, and received us every one, because of the present rain, and because of the cold.
 And when Paul had gathered a bundle of sticks, and laid them on the fire, there came a viper out of the heat, and fastened on his hand.
 And when the barbarians saw the venomous beast hang on his hand, they said among themselves, No doubt this man is a murderer, whom, though he has escaped the sea, yet vengeance suffers not to live.
 And he shook off the beast into the fire, and felt no harm.
 However, they looked when he should have swollen, or fallen down dead suddenly: but after they had looked a great while, and saw no harm come to him, they changed their minds, and said that he was a god.
American King James Version×).
Additionally, the Bible tells us that we should not tempt God by deliberately placing ourselves in potential danger (Matthew 4:5 Matthew 4:5Then the devil takes him up into the holy city, and sets him on a pinnacle of the temple,
American King James Version×, 7). Based on these scriptures, we can conclude that God does not want us to handle poisonous snakes today to prove we are Christians. We can also be assured that God continues to perform miracles in healing and protecting those who trust and believe in Him.