The first thing you need to do is a complete DeOxit cleaning (see the sticky - I sure ain't typing all that over again!
). Make sure you check/clean everything mentioned there. This will eliminate any problems that arise from the first and most common source of sound degradation and noise: dirty contact surfaces.
While you are in there, take a look at the caps. Any with bulging ends or opened seams or stuff visibly leaking out of them MUST be replaced, of course. Don't confuse glue-like stuff used to anchor some large caps against vibration with stuff leaking out: look carefully to see the difference.
Check DC offset. If you have more DC coming out than you should, it's probably due to aging caps, and can be a sign that a recap might be in order. IF it's only a wee little bit more than you'd like, you don't have to rush, but if it is well out-of-spec, I'd take it as a good indicator that it's time to get out the soldering iron.
After that, fire it up and give the sound a good listen. Do you hear any strange noises? Anything that sounds like a helicopter, going whoop-whoop in rhythmic cycles? Anything sizzling like frying bacon? If so, then you have some SERIOUS cap issues with at least one cap entering its final-stage death throes, and it's only a matter of time (probably not long) before it gives up the ghost entirely. If you hear those kind of noises, DO NOT use the gear until you can do a recap. Otherwise, you risk taking out other, and perhaps costly or hard-to-find, components inside.
With other forms of sonic degradation, it can be hard to identify the cause without some experience and/or testing. In general, sound will become thinner, more brittle-sounding, more compressed and less dynamic, less full, less full-range, etc... as caps age. The background noise level ("noise floor") will also increase, making details less distinct, soundstage less clear, etc... Your gear will begin to sound less like hi-fi stuff and more like an old transistor radio, if that helps describe it.
The problem here is that other things can also contribute to deteriorated sound. For example, some old carbon resistors get noisier as they age, and contribute to a less silent background. However, most of the time, the bulk of sound deterioration is due to aging caps. Capacitors by their nature dry out and sonically deteriorate over time, more than most other components. If you have vintage gear that is over 20 years old and has not been recapped, the odds are high that it is due for a recap, and will sound MUCH better once the caps have been replaced. You can use an ESR meter to get a good estimate of how "aged" the caps are, but in general as soon as some start to fail (or get close to failure), it's probably best to replace all of the ones of similar age. Not everyone agrees with this approach, but I think it makes the most sense.
New caps are generally better made than they used to be, as cap-manufacturing technology has advanced over the years, and you may decide to use ones with better characteristics (a complicated subject, as what is "better" will depend on the position/function in the circuit), tighter tolerances, etc... to "upgrade" the sound.
As long as you are (or your tech is) in there, check all the other components, too: re-mount transistors with fresh thermal grease (Important! The stuff dries out, just as capacitors do.), check resistors for being in spec and if they are old carbon types, replace them, etc... Set bias and DC offset if appropriate; get a proper alignment if it's a tuner (not a job for amateurs; send it to an expert), and you should be worry-free for years to come...
...until the contacts get dirty and the new caps start becoming old caps....
