While the New Age movement has commandeered the term, I do think Christianity has a mystical aspect to it. I believe I got the following analogy from C.S. Lewis.
Discovering spirituality is like discovering you are in a boat. Mysticism is like pushing off from the dock. Since many leave safe mooring and perish in the waves, this is not to be done in a cavalier fashion - even though it can be exciting to push off into the deep.
The issue is not of whether we should push off, for Christians must do so as well if they intend to get anywhere (and that is what boats are for), but rather of where you are going.
The non-Christian Mystic is going for the experience itself, an aimless ride, and would be better off in the long-run clinging to the dock. The Christian casts off from this world as well, but with clear intent to where he is headed, with the best of maps, circumspectly, deliberately. The ride may be similar, but the end effect wholly different. The Christian Mystic arrives, against all dangers and odds. The non-Christian Mystic drifts aimlessly, and eventually starves or drowns.
Thus we launch out with fear and trembling, but trust that He who commanded us to do so can calm the waves, and see us through to His real, safe port.
1 Tim 3:16 (NIV) Beyond all question, the mystery of godliness is great.
Col 1:27 (NIV) ...God has chosen to make known... the glorious riches of this mystery, which is Christ in you, the hope of glory.
John 20:29b (NIV) "Blessed are those who have not seen and yet have believed."