A realistic way Christians can always judge whether they should do something, such as deciding to vote or not, is to ask what Jesus would do. The apostle John once wrote, "Anyone who claims to dwell in Him (Jesus) is obligating himself also to walk even as He Himself walked" (1John 2:6, HBFV throughout). The "what would Jesus do" admonishment is solidly based on Scripture and bears on the issue of casting a vote.
So, would Jesus vote? Moreover, if He did, whom would He choose? Would he choose a Christian? Would He be willing to choose the "lesser evil" of whoever is running for office? On the other hand, however, would He stand apart from the world and proclaim a different way? In should be noted that the Bible does not record Christ every casting any vote in anything related to the Roman empire.
On trial for His life, Jesus told Roman Prefect Pontius Pilate the following when asked whether he was the King of the Jews. He stated, "My kingdom is not of this world. If My kingdom were of this world, then would My servants fight, so that I might not be delivered up to the Jews. However, My kingdom is not of this world" (John 18:36, HBFV).
Jesus proclaimed that His followers would not fight to save Him from an unjust execution because His kingdom (God's government) was not actively ruling over the world. Given this fact, why should Christians be so eager to get involved in politics such as trying to vote? This world is clearly deceived by Satan and lies under his power and sway (at least for the time being, see Revelation 12:9, 1John 5:19, Matthew 4:8, and so on).
Human governments, whether or not they allow their citizens to vote, cannot get rid of evil human nature, Satan's influence, or the corruption of this world's deceived civilizations. At his Second Coming, however, Christ will set up the kingdom of God on earth that will solve all the problems that human governments are incapable of resolving.
How are leaders chosen?
We should not believe that the duties or rights of earthly citizenship require Christians to participate in its political process such as choosing who rules a nation or sets its laws. Our vote does not guarantee certain people will be placed in positions of power. It is God who sets up and removes leaders (Daniel 2:20). When He made Nebuchadnezzar of Babylon insane for seven years, the reason given for the punishment was to teach him that the Eternal rules over the kingdoms of men (Daniel 4:17).
What can we do?
Some politically related activity, unrelated to casting a vote, might be fine such as writing letters to the editor of a newspaper condemning sinful activities or voicing our opinion on laws that would censor true believers. If a church is being publicly persecuted, it would be good for its members to protest those unjust laws publicly, such as when a government jails a church's leaders or members for obeying God instead of men (Acts 4:19, 5:29).
Should Christians vote? Believers ought not to waste their time and energies on trying to fix a dying world through the ballot box. Our heavenly Father wants us to dedicate ourselves solely to worshipping Him in spirit and truth, instead of incrementally trying to improve a system Satan dominates (2Corinthians 4:4). Our goal should be to proclaim to the world what are God's solutions to man's problems.