We have long speculated, and were finally shown in ST:Enterprise, that the "cheat" that the starships use creates some sort of a sharpish border between the universe and the small subsection of the universe that immediately surrounds the ship. That is, the "warp field" extends to a certain distance from the ship, and the laws of nature inside it and outside it are at a major disagreement.
What happens if, say, your head gets caught in the field as the ship passes by? Why, it will get smeared across several light-seconds by being grabbed by the warp field. Or more probably, by the outer layers of the onionlike, nested warp fields, with successive greater degrees of disagreement with the universe on the subject of how fast things are really moving. The end result in any case is messy decapitation, or the same thing that happens to this bit of cable that strays outside the warp field of the two starships in this scene from ENT "Divergence":
I guess another interesting question is, what will you see? Theoretically, the ship will be within your field of vision for a few milliseconds: light from its closest passing will hit your eyes first, then light from it being a meter ahead of you and
behind you, then from two meters ahead and
behind, and so forth. So you will see a brief flash that then splits in two and goes away from you in both the directions of arrival and departure...
...That is, if anything at all will be visible. Somehow, the warp field stops the starship's immediate vicinity and the rest of the universe from talking to each other. It might well prevent an outside observer from seeing anything much, too - perhaps the "cameras" that "record" the scenes we see in the episodes are mere simulations of what would be seen if seeing were possible, or perhaps they themselves are placed inside the warp fields, or something.