Rules as Written
Consider the case where the caster readies, as Marq says, to "blast the first thing that comes through the door". With that as their trigger, the caster gets only one chance. When the first thing comes through the door, they can react to the trigger or ignore it. If they ignore it, however, they have now lost their readied action, because there will be no other first thing through the door - that trigger cannot repeat (at least not this turn).
However, suppose a more careful caster worded their trigger as readying to "blast a foe when it comes through the door". This trigger could be repeated multiple times before their next turn - if they ignore the first creature, a lowly xvart, are they allowed to use their reaction at any future time when a foe comes through the door?
Unfortunately, RAW does not answer this. We are told "When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger." We know that if you take your reaction, you cannot respond to a subsequent trigger, because you get only one reaction per turn. But when you ignore the trigger, we are not told what happens - the rules do not specify whether you can then hold your reaction for a subsequent trigger or whether in ignoring the trigger your reaction is equally as spent as if you had used it1.
Let's compare it to other reactions
Since there is no clear indication from the text for how this is supposed to work, we can cast a wider net and look at how similar mechanics work. Readied actions use your reaction, and there are other things that use reactions as well, things that might be said to have 'automatic triggers' or 'repeated triggers'. For example, whenever an opponent you can see leaves your reach, you are permitted to make an opportunity attack as a reaction. If you do so, your reaction is used. However, if you do not choose to do so, your reaction is not used. The next time (and the next and the next) this happens, you are permitted to make the choice again. Ignoring the first opportunity does not remove your ability to react to any future opportunities.
Suppose a player told you, "As my action, I want to ready an action to attack an opponent that leaves my reach." You might respond, "That's an opportunity attack, you get that anyway." To which they answer, "Nevertheless, it is a reaction to a perceivable circumstance and a legal readied action," and you accept it. Then the first time an opponent leaves their reach they ignore the trigger, waiting for another opponent. Do you then disallow all future opportunity attacks? "I'm sorry, once you said it was readied, that locks you into taking it either the first time or never - you should have not readied it so that you would be able to do it at any time." Does this make sense?
And it turns out that this is the general rule for reactions. A caster with a reaction available when hit by an attack may choose to cast shield as a reaction, and that choice may be made any time they are hit - ignoring the first triggering event does not prevent them from using the spell later in the round. An enchanter with instinctive charm can use their reaction whenever a creature within 30 feet attacks them - even if they have ignored previous creatures that have done so. By and large, class features, spells, and other powers that require the use of a reaction are flexible in terms of their use and do not require that you 'use it or lose it' at the first opportunity. Why should readied actions be any different?
As a general rule, it is good DM practice to encourage player agency and give players more choices, rather than fewer. If a player has already forgone their action on their turn and invested in a readied action, they are likely to feel cheated if their use of that action is even further restricted. Consider a caster who readied an action to 'blast anything that comes through the door'. The DM has a goblin pick up a rock, throw it through the door, and then enter. Even if the DM had planned to have the goblin do this before the player announced their intent, the player is likely to feel cheated by a 'gotcha DM' who then says, "Well, you can blast the rock or ignore it but either way you can't attack the goblin." A player then trying to avoid such a situation falls into saying things like "I ready an action to attack the first creature that comes through the door, provided I think it is an enemy, is susceptible to my attack, and my attack is unlikely to do more than twice its maximum hp." This quickly becomes an adversarial situation where the DM and player are conflicting through words and definitions rather than fantasy characters, and is ultimately unlikely to be satisfying to either of them. In my experience, it is better to generously allow players options for their actions, and then apply strictly realistic resolutions to those actions, than to try to restrict their options for actions even before they choose. With more options, even when things do not go as they intended, they still feel satisfied that they have been able to make the choices they wanted.
1Jeremy Crawford has tweeted in favor of you retaining your readied action for a subsequent trigger, but this is not official:
Ready action. The rules says you can ignore the trigger, because yes, you can ignore it. You can try again if it is open-ended enough.