Being "carnally minded" is placing one's faith in something other than Christ and the Cross, i.e., The Finished Work, i.e., The Blood of Jesus.
Being "spiritually minded" is what one must be to successfully "walk after the Spirit." It simply means placing one's Faith exclusively in Christ and the Cross, excluding all else, and not allowing anything to take the place of the Cross.
Paul uses the terms, "the flesh" or "in the flesh," quite often. What does he mean? He is speaking of a person's own natural ability, our natural strength, our own efforts, in other words, what we as human being can do.
We cannot do anything within our own strength and power that will gain anything with the Lord, no matter how noble our efforts might be. In other words, the Lord cannot accept our efforts, our ability, our personal power, etc. So to have what we must receive from the Lord, that is, if we are to live as we ought to live, then we will have to depend totally, and I mean totally, on Christ and what Christ has done for us at the Cross.
To help us understand it better, anything we try to do, any way we attempt to go, any effort we attempt to make, other than by Faith in Christ and the Cross, is labeled by the Holy Spirit as "the flesh," which is displeasing to God, and which will never bring victory.