Setting and Overview
Matthew 10:16 serves as something of a transitional verse between the content of 10:1-15, which seems to focus primarily on the immediate future, and verses 17 and on which seem to focus on a longer time horizon. It is a very effective transition, as well, capping the previous verses and setting up the following with a dramatic statement, certain to fully capture the attention of the disciples with the vividness of its warnings, and powerful enough to continue in the centuries to come as a timeless warning and exhortation to all followers of Jesus.

Exposition
When Jesus refers to sheep amongst wolves, a warning repeated in the sending of the seventy-two (Luke 10:3), and likely other occasions as well, he is employing an illustration that carries significance to those familiar with sheep and those familiar with the literary significance of the term. In the Old Testament, sheep are often used to refer to the dependency of man (Psalm 23), and especially man’s tendency to be helpless and led astray (Ezekiel 34.) Importantly, a sheep has virtually no offensive capability, meaning that it is susceptible to violent intentions. That connotation appears frequently in intertestamental literature, both emphasizing the vulnerability of Israel, and thus the sovereignty and faithfulness of God when He delivers them. Unlike in Luke 10:3, the focus here is not as much on their dependence on God (although that is certainly assumed still) but rather on their need to be careful. A sheep’s only hope to survive on its own is to avoid unnecessary danger, and they must be as mindful as a helpless sheep among wolves.
How they are supposed to do that begins with the wisdom of a serpent. The most common word for wisdom in the New Testament is sophia, which has a broad meaning of intelligence or knowledge. The word used here for wisdom is the other word, phronimos, which focuses more on wisdom as lived out with thoughtfulness or prudence, and is used elsewhere in Matthew to emphasize forethought (7:24, 24:45, 25:1-13.) Although the latter word isn’t used in application much elsewhere in the New Testament, we do still get a clear picture of what that looks like when Paul talks about thoughtfully using sophia (Colossians 4:5-6, I Corinthians 14:20, Ephesians 5:15-16, Romans 16:17-19.)
In addition, we add the innocence of a dove. The word for innocence, literally meaning to be pure or unmixed, is only elsewhere in the New Testament in Philippians 2:14-16 and Romans 16:19. The imagery there of innocent children challenges us further to live not according to our old identities, or even to be influenced from our former bitterness or habits, but rather to live out wisdom with the gentleness that can only come with maturity in Christ (Ephesians 4:32, James 3:13.)
Taken alone, the wisdom of the serpent connotes, along with the positive aspects seen previously, also a shrewdness (Genesis 3:1, Luke 16:8) that is usually associated with negative aspects such as cynicism, malice, manipulation, and insensitivity. Likewise, the innocence of a dove adds to the positive aspects the idea of weakness, gullibility, naivete, and carelessness (Hosea 7:11.) However, when put together, the positive aspects of each eliminate the possibility of the negative of the either, exhorting us to live out the positive aspects of both fully and consistently at all times.

Examination and Application
As we also live as sheep among wolves, we must still not permit ourselves to only live out half at a time. If we are only as wise as serpents in the face of opposition, then we are likely to push people away in need of grace and hope. If we are only as innocent as doves, then we are likely to be mislead and to be ineffective in our ministry. If we are both together, then we can face such opposition with the vigilance, but also with gentleness and patience (I Peter 3:15-16, II Timothy 2:24-25.)