A 12-volt alternator draws about 4 amps when the master and ALT switches are on and the engine isn't running. That's a minimal draw, but if one happens to forget to turn stuff off after shutdown, that four amps slowly heats up the rotor's field winding and can roast it. I have seen that only once.
The alternator will take a bit of energy to turn during start, increasing the load on the starter a little bit. When starting might be a marginal proposition, such as in cold weather, better to leave it off, then turn it on once started. The risk here is forgetting to to turn it on. Keeping the battery charged up while parked is fine, but starting the engine eats a whole pile of power and the alternator has to replace all that anyway.
It does take a bit more for the alternator to recharge a low battery, but a low battery also won't start the airplane too well and most owners will replace it.
The alternator's job is to recharge the battery after start, and to supply all the airplane's needs during that recharge and after the recharge. In the typical lightplane, a 60-amp alternator has no trouble whatever doing this. It is designed to handle a whole 60 amps continuously. Where you might see some problems is at night, with all the lights on, and taxiing endlessly. Low RPM means high field current (that four amps), high output, and poor cooling, as the fan isn't pulling much air through the alternator. Still, alternators are a hundred times better than the old generators were. We should be grateful for them.
Most alternator failures are due to worn-out brushes. It's false economy to let them get that way, since a worn brush falls out of its holder and the spring behind it then arcs on the slip ring and chews it up and burns it out. Now you're not buying new brushes ($20), you're buying a new alternator. $500?
And a failed alternator can kill you if it's at night or in IMC and you have some distance to go. Battery dies, dark cockpit, no radios, no nothing. Does that sound like fun? Shouldn't that alternator get some preventive maintenance?
Some rebuilders are in the habit of putting far too much grease in the rear bearing. That grease squeezes out when the alternator is assembled and the rotor shaft goes into the bearing, and then it ends up on the slip rings, sludging them and the brushes up and causing electrical resistance that limits the field current and therefore the alternator's output. Kelly/Hartzell probably still haven't learned that lesson even after 20 years. I filed way too many SDRs on it for them to have any excuse. Some independent rebuilders are just as bad.