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
(as in Romans 8.)    
Abingdon's Strong's
Exhaustive Concordance of the Bible
gives us additional insight on the word 'Walk': 
The word as it it 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).
This I say then, Walk in the Spirit,
and ye shall not fulfill
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: 
A 'Votary' is
"one devoted , consecrated,
or engaged by a vow
or promise".