It depends on how your escalator is running.
It seems like you want the simplest thing possible, and it seems to me the simplest thing is an escalator whose mechanism can only apply a single, constant, force/torque (I say force/torque because we aren't being specific about what is actually causing the escalator to move. From now on I will just say force.).
Let's consider before anyone gets on the escalator. It is supplying enough force to move the mass of the escalator parts as well as resist internal frictions that are present (which for our discussion we will assume to be constant friction forces that do not depend on the velocity of the escalator. Having velocity dependent friction could change some things, but I will leave thinking about that to you). Let's assume at this point the stairs are moving at a constant speed.
Now someone steps on. If we want to maintain a constant speed up the escalator, the thing supplying our force would need to increase its "strength" to counter the new "resistance" caused by the person stepping on. It needs to be able to lift the stairs, fight the friction, and now also lift the person. Since our force being supplied is constant, it cannot do this. Therefore the escalator would actually begin to slow down, and then eventually start to move backwards if the person does not get off in time (this could change with a velocity dependent friction force). This would be analogous to me lifting my hand at a constant speed, and then someone puts a mass in my hand while I supply the same upward force to my hand. The net force on the mass will be downwards, so it will have a downward acceleration.
If something is keeping the speed from changing, does that mean the torque of the motor changes? Or does it simply draw more power but keep the same torque?
So let's say our escalator can adjust its applied force/torque accordingly to keep the escalator moving at a constant rate. Then yes, the force would need to increase to keep people moving at the constant rate. Just how if someone puts that mass onto my hand I will need to push harder to keep my hand (and mass) moving at a constant speed. This also means more power is needed too. If we want the escalator to move at a constant rate, but it is doing more work to lift the people up, then it is doing more work in the same amount of time. Therefore, the rate of energy expenditure increases, i.e. more power needs to be supplied.
Therefore, to answer your question title, escalators don't slow down because they are designed to be able to rotate the steps at a constant speed by adjusting the supplied force/torque.