Paul had contrasted life in the flesh to life in the Spirit. He continued in this theme.
Romans 8:12-13 (KJV)
Therefore, brethren, we are debtors, not to the flesh, to live after the flesh.
For if ye live after the flesh, ye shall die: but if ye through the Spirit do mortify the deeds of the body, ye shall live.
According to verse 12 believers are debtors.
1 Corinthians 6:19-20 (KJV)
What? know ye not that your body is the temple of the Holy Ghost which is in you, which ye have of God, and ye are not your own?
For ye are bought with a price: therefore glorify God in your body, and in your spirit, which are God's.
They are bought with a price. It took the Creator God to offer himself as a sacrifice for the salvation of those who would believe. He took the penalty that sinful man deserved and was crucified. Through Him believers have eternal life. They owe Him everything.
Yet, believers are not debtors to the body with its fleshly desires. For if believers lived according to those desires, they would die (verse 13). Instead the believers were, through the power of the indwelling Holy Spirit, to mortify (bring under control) the deeds of the body. Then they would live.
It is important to understand that the desires of the flesh refers to those attractions and pulls that lead to sin, and are not those desires that are acceptable to God. In other words caring for your body is not what is in mind here. Throughout church history believers have attempted to gain God's approval by some rather horrendous practices, such as climbing stairs on the knees, refusing food, and abusing the body. This was never God's intent.
Yet, these two verses pack a punch! Without question, salvation comes by grace through faith. However, these verses clearly indicate that continuing to live according to the desires of the flesh is inconsistent with true belief. When the Holy Spirit lives within believers, desires and behaviors change.
This behooves us to examine ourselves carefully! If we are living in habitual sin, or have the idea that God accepts us in our sinful state and we do not change, we are likely not saved!