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This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.

141 "Whoah! Hold on! They must be moving faster - look at the stars that shoot past while they're in warp!" (Warp Velocities - Star Trek)

Joseph Haller offers:

The most extreme ship induced speed discussed ... is W(ST:TNG) =
9.97535, or 788,940c.

This would give a characteristic angular speed for nearby stars of
1578 arc-seconds per second or 1 degree every 2.3 seconds. This is
indeed verified in the simulations. Travel at high warp speeds, on
the ST:TNG warp scale, does not match very well the appearance of
the bridge view screen on a typical episode. Indeed, most visible
stars are not nearby but are further away with correspondingly
lower angular speeds. I offer no solutions to this discrepancy
other than the dramatic necessity that stars go whooshing by at
high warp speed.

Or should we give up so easily?

There's a lot of support on rec.arts.startrek.tech for the notion that
those things aren't really stars. For one, as the Enterprise drops out
of warp (with the camera tagging along for the ride) some of the
"stars" do some pretty strange things, such as suddenly angling off in
various directions, disappearing, etc.

Also, in Star Trek: First Contact, the Phoenix barely breaks Warp 1
and stays relatively close to Earth, but we still see the streaks.
Definitely not stars.

The predominant theory is that what we're seeing are free particles in
space interacting with the expanding boundaries of the warp field. As
they cross the warp field, they are repeatedly accelerated to FTL
velocities and then slowed to STL speeds, and start spewing out
something like Cerenkov radiation, a (real!) bluish light emitted when
particles moving faster than the local speed of light (in a dense
medium) are forced to slow down. If not exactly Cerenkov radiation,
then something similar.

Jon Mitchell tells me that in the TNG video game for the Sega Genesis
console platform states the streaks are part of the visual
manifestation of Einsteinian space in subspace. So people other than
us .techers have noticed this problem too.

As a side note, in "The Cage" [TOS], the moving particles seen through
the forward view-screen are explicitly identified as meteoroids.

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