Degaussing should be the first thing attempted whenever color purity problems are detected. As noted below, first try the internal degauss circuits of the TV or monitor by power cycling a few times (on for a minute, off for at least 20 minutes, on for a minute, etc.) If this does not help or does not completely cure the problem, then you can try manually degaussing.
Note: Some monitors have a degauss button, and monitors and TVs that are microprocessor controlled may degauss automatically upon power-on (but may require pulling the plug to do a hard reset) regardless of the amount of off time. However, repeated use of these 'features' in rapid succession may result in overheating of the degauss coil or other components. The 20 minutes off/1 minute on precedure is guaranteed to be safe. (Some others may degauss upon power-on as long as the previous degauss was not done within some predetermined amount of time - they keep track with an internal timer.)
Commercial CRT Degaussers are available from parts distributors like MCM Electronics and consist of a hundred or so turns of magnet wire in a 6-12 inch coil. They include a line cord and momentary switch. You flip on the switch, and bring the coil to within several inches of the screen face. Then you slowly draw the center of the coil toward one edge of the screen and trace the perimeter of the screen face. Then return to the original position of the coil being flat against the center of the screen. Next, slowly decrease the field to zero by backing straight up across the room as you hold the coil. When you are farther than 5 feet away you can release the line switch.
The key word here is ** slow **. Go too fast and you will freeze the instantaneous intensity of the 50/60 Hz AC magnetic field variation into the ferrous components of the CRT and may make the problem worse.
WARNING: Don't attempt to degauss inside or in the back of the set (near the CRT neck. This can demagnetize the relatively weak purity and convergence magnets which may turn a simple repair into a feature length extravaganza!
It looks really cool to do this while the CRT is powered. The kids will love the color effects (but then lock your degaussing coil safely away so they don't try it on every TV and monitor in the house!).
Bulk tape erasers, tape head degaussers, open frame transformers, and the "butt-end" of a weller soldering gun can be used as CRT demagnetizers but it just takes a little longer. (Be careful not to scratch the screen face with anything sharp. For the Weller, the tip needs to be in place to get enough magnetic field.) It is imperative to have the CRT running when using these whimpier approaches, so that you can see where there are still impurities. Never release the power switch until you're 4 or 5 feet away from the screen or you'll have to start over.
I've never known of anything being damaged by excess manual degaussing as long as you don't attempt to degauss INSIDE or the back of the set - it is possible to demagnetize geometry correction, purity, and static converence magnets in the process! However, I would recommend keeping really powerful bulk tape erasers-turned-degaussers a couple of inches from the CRT.
Another alternative which has been known to work is to place another similar size monitor face-to-face with the suspect monitor (take care not to bump or scratch the screens!) and activate degauss function on the working monitor. While not ideal, this may be enough to also degauss the broken one.
If an AC degaussing coil or substitute is unavailable, I have even done degaussed with a permanent magnet but this is not recommended since it is more likely to make the problem worse than better. However, if the display is unusable as is, then using a small magnet can do no harm. (Don't use a 20 pound speaker or magnetron magnet as you may rip the shadow mask right out of the CRT - well at least distort it beyond repair. What I have in mind is something about as powerful as a refrigerator magnet.)
Keep degaussing fields away from magnetic media. It is a good idea to avoid degaussing in a room with floppies or back-up tapes. When removing media from a room remember to check desk drawers and manuals for stray floppies, too.
It is unlikely that you could actually affect magnetic media but better safe than sorry. Of the devices mentioned above, only a bulk eraser or strong permanent magnet are likely to have any effect - and then only when at extremely close range (direct contact with media container).
All color CRTs include a built-in degaussing coil wrapped around the perimeter of the CRT face. These are activated each time the CRT is powered up cold by a 3 terminal thermister device or other control circuitry. This is why it is often suggested that color purity problems may go away "in a few days". It isn't a matter of time; it's the number of cold power ups that causes it. It takes about 15 minutes of the power being off for each cool down cycle. These built-in coils with thermal control are never as effective as external coils.
Note that while the monochrome CRTs used in B/W and projection TVs and mono monitors don't have anything inside to get magnetized, the chassis or other cabinet parts of the equipment may still need degaussing. While this isn't likely from normal use or even after being moved or reoriented, a powerful magnet (like that from a large speaker) could leave iron, steel, or other ferrous parts with enough residual magnetism to cause a noticeable problem.
See the document: TV and Monitor CRT (Picture Tube) Information for some additional discussion of degaussing tools, techniques, treatments for severe magnetization from lightning strikes, and cautions.