A very good question. One that requires several long answers.
First, your question directly asks what audible things can we hear if an old crossover has gone bad. There are several possible things that may go wrong with an old, poorly functioning crossover. But there is no single thing that stands out. It depends on the individual drivers and the original intent of the crossover. You might hear noise from a woofer in break-up mode, or you might hear a gap between a woofer and tweeter. The break up noise depends on the music played, but it usually sounds harsh and prominent. A gap between the woofer and tweeter might easily be overlooked. All this is best heard if you can directly compare two speakers identical except for their crossovers.
My second answer is to a question you didn't directly ask, 'what does a poorly functioning or poorly designed crossover sound like?' Underlying this is the fact that many commercially available speakers, when new, come with poorly designed crossovers. I know one particular DIY speaker designer who made his excellent reputation because of this. (I'd like to able to say he made a good living because of this, but that's another story.)
Now is a good time to refer you to
this web page, where Dennis Murphy discusses some fundamental issues in crossover design.
The other question you didn't ask is about crossover components themselves, some of which may or may not age and gradually go bad. In general, passive crossovers are made with 3 different types of crossover components, inductors, resistors, and capacitors. Inductors (also known as chokes or coils) and resistors rarely go bad and are not known to age. On the other hand, some capacitors can fail gradually with age or more quicker due to faulty construction. Much of the chatter on internet audio forums about aged crossovers is really about how some very cheap capacitors have been known to go out of spec or fail outright with age. Some of that chatter is correct, but much of it takes some simple facts and pushes them beyond reason.
I don't have enough time, space, or patience to go into a treatise on the different types of capacitors.
Wikipedia is a good source of info, and that should keep you busy reading it for some time. A simple summary, as I understand things, for capacitors that appear in most or all audio crossovers appears below:
There are basically 2 types of capacitors, electrolytic and film. Electrolytic caps, often non-polar electrolytic (NPE) caps usually cost much less that the various film types, especially in the past. Some, not all, were so poorly made that many were out of spec when new. In the past 20 years or so, less expensive film caps have become widely available. They seem to be well made, and may never drift out of spec, or fail. To borrow a joke from Dennis, I'll be out of spec long before the film caps in my speakers will be.
Some NPE caps have been known to fail with age, due to drying out, or due to poor manufacturing quality. Again,
see Wikipedia for details, especially a page titled Capacitor Plague.
In my opinion, much of the internet chatter in audio forums about aged capacitors comes from NPE caps in older speakers. Some of it may be correct, but how much of it comes because of the so-called capacitor plague? And how much of it came because of poor crossover design when the speakers were new?
Again, the usual caveat, I'm self-taught. I'm not an electrical engineer and I didn't cut my teeth in school learning about these things with proper academic rigor. All this talk about capacitors is limited to those usually used in building passive audio crossovers. There are many other uses of capacitors, but I'm not discussing them here.