as it is used in this verse, means,

"To live and act or behave; to pursue a particular course of life."
Webster's 1828

Webster states that to 'Walk after the Spirit' is "To be guided by the counsels and influences of the Spirit and by the Word of God, and to live a life of holy deportment."
(as in Romans 8)

This I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh. For the flesh lusteth against the Spirit, and the Spirit against the flesh: and these are contrary the one to the other: so that ye cannot do the things that ye would. But if ye be led of the Spirit, ye are not under the law.
Galatians 5:16-18

Abingdon's Strong's Exhaustive Concordance of the Bible gives us additional insight on the word 'Walk':
The word as it is used in the above verse is translated from a Greek word meaning, "...to tread all around, i.e. walk at large (especially as proof of ability); figuratively to live, deport oneself, follow (as a companion or votary):-- go, be occupied with, walk (about).

A 'Votary' is "one devoted, consecrated, or engaged by a vow or promise".
Webster's 1828