I'm playing a Fender Jaguar, and when I switch from the bridge to neck humbucker there is a noticeable volume difference. The neck pickup gives me a funky clean tone which I like, and the bridge gives me a more heavy bright kind of tone.

I would like to either raise the neck pickup or lower the bridge pickup to get the volume levels more even. Which would be the best to do, and would doing this alter the tone greatly?

Would pictures of the string level help you at all?

Yes changing the pickup height will greatly alter your tone. The farther away the pickup is the more quiet it will be, obviously, but the tone will also get clearer. It's the opposite for too close. Remember to adjust the pickup height screws a quarter turn at a time. You will be surprised at how much difference a quarter turn in either direction makes.
Keep in mind, if you get the pickups too close to the strings, the magnets in the pickups will have the effect of dampening string vibrations. This will result in a loss of sustain. This does not apply to active pickups, which can be as close to the strings as you can get them.
Thanks for the tips. Do you suggest moving the neck closer or bringing the bridge down?

edit: Is it safe to do while the guitar is plugged in?
You can move the pickups in any direction you like until the desired tone is achieved. I did my pickup adjustment with it plugged in just so I could make the adjustment then play it for a little while and make another adjustment if necessary. I'm still alive so it must be safe, but if you want to be extra safe, unplug it each time you make adjustments. Remember that these settings aren't set in stone, so if later down the line you still don't like the tone you can adjust them again. There's not really any right or wrong way to adjust pickup height.
Unless your pickups are active, your guitar is passive, therefore there is no electricity running through it. Plugging it into your amp won't transfer current from your amp into your guitar, and 9v isn't really anything substantial to be worried about even if it is active.
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Pickup height does make a big difference. I usually have my bridge closer then my neck. But the bridge isn't very close to the strings though. Level would be a good start.
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The distance between the top of the pickup and the strings is usually twice as big at the neck as it is at the bridge. This does, of course, depend on which pickups you are using and the way you want your pickups voiced. Putting the pickup closer to the strings gives you better top end and more volume. Moving it further from the strings helps to fill out the low end and gives you a more open and dynamic tone but less volume.
I now have my neck pickup slightly lowered than the bridge. The neck pickup still gives a warm tone whilst the bridge gives a brighter punchier sound, but the levels are more or less equal.

The neck pickup was wayyy too low down, virtually in the guitar body, and the bridge was nearly touching the strings. Nice one fender.