<p>I've heard I should store filament in sealed container, preferably with a desiccant. </p>

<p>But let's say I let a spool get a little old on the printer, or I purchased a filament spool that was old or improperly packaged. How would I know? How would this impact prints (what kind of symptoms would I see)? What things could I do (perhaps in the slicer) to correct for this and prolong the life of marginal filament?</p>

<p>And the corollary... in a typical environment, how long can filament be left out without suffering from the exposure?</p>

<p>I'm thinking mainly about PLA, but responses for ABS and other materials are useful, too.</p>

The primary issue with long-term exposure of filament to the environment is that it will absorb water moisture from the air. When a filament that has absorbed water is passing though the hot end of a printer, the water will turn to steam and cause problems with extrusion:
Small bubbles of steam can form, causing extrusion to sputter - you might hear a sizzling noise and have poor consistency.
Large steam bubbles can cause significant oozing followed by no extrusion.
Extreme cases can cause mysterious jams that seem to clear themselves (the extruder cannot overcome the steam pressure).
In short, this will cause terrible print quality and failed prints. As the effects are not consistent, there is nothing that can be done by slicer settings to "recalibrate" for filament that has absorbed water.
This can be avoided by storing filament in an air-tight container with desiccant to ensure low humidity. Some people use "dry boxes" that allow the spool to be mounted inside while filament can be passed to the printer, so there is minimal exposure even while the spool is in use.
If you do suspect that your filament has absorbed moisture, you can dry it out, by placing the spool in a warm oven or in a food dehydrator for a few hours. If you weight it before and after, you should find that it weighs several grams less afterwards.
WARNING:
It is important that the temperature does not soften the plastic at all, or it can become distorted or bind on the spool. Most ovens will peak well above the set temperature as the thermostat cycles. Of course, fully melting a roll of filament could destroy your oven or cause a fire.
It's hard to say how much environmental exposure is too much, as every filament and environment is different. When I started out, I had several spools of PLA that I stored in the open for months. I didn't think I was having any problems, but I was also learning much and improving my printer settings at the same time. After getting PETG, it became unusable with oozing and jams after about two weeks but a few hours in my oven was a miracle cure! I then dried some PLA as well, and I found that print quality did improve, but not amazingly so. I have not used ABS, but in theory it is less hygroscopic than PLA, so it is probably not very sensitive to exposure.
I set up a dry storage box, and I am careful to always store PETG or my "good" PLA when I'm not actively using it. I have a couple rolls of PLA that I don't like as much anyway and generally just use for draft prints, and I don't really worry about it that much.
Note: An object that has been printed will also absorb moisture, but in general this isn't a problem.