I was verifying whether my fighter's Sun Blade would in fact be a useless hilt in an antimagic field (AMF) and came across this question on a few other sites, and it piqued my interest.
My first instinct is that "no, monks can still use ki effects in an AMF because they're just awesomely in control of their bodies" ... but then I thought about the monks that nobody plays. I can't see a monk being able to cast burning hands or any other Way of the Four Elements spells in an AMF.
Antimagic Field (PHB, p. 213)
A 10-foot-radius invisible sphere of antimagic surrounds you. This area is divorced from the magical energy that suffuses the multiverse. Within the sphere, spells can't be cast, summoned creatures disappear, and even magic items become mundane. Until the spell ends, the sphere moves with you, centered on you. Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
Targeted Effects. Spells and other magical effects, such as magic missile and charm person, that target a creature or an object in the sphere have no effect on that target.
Argument for ki abilities being negated by AMF:
The Magic of Ki (PHB, p. 76)
Monks make careful study of a magical energy that most monastic traditions call ki. This energy is an element of the magic that suffuses the multiverse-specifically, the element that flows through living bodies. Monks harness this power within themselves to create magical effects and exceed their bodies' physical capabilities, and some of their special attacks can hinder the flow of ki in their opponents. Using this energy, monks channel uncanny speed and strength into their unarmed strikes. As they gain experience, their martial training and their mastery of ki gives them more power over their bodies and the bodies of their foes.
Argument against ki abilities being negated by AMF:
Ki-Empowered Strikes (PHB, p. 79)
Starting at 6th level, your unarmed strikes count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
It specifically doesn't say that being Ki-Empowered is magical, leading me to believe that it isn't magical. Therefore, it wouldn't be negated in an AMF.
Way of the Four Elements (PHB, p. 80)
You follow a monastic tradition that teaches you to harness the elements. When you focus your ki, you can align yourself with the forces of creation and bend the four elements to your will, using them as an extension of your body...
Again, it doesn't say anything about interacting with the Weave or making a genuinely magical effect. That said, it also doesn't say that the element needs to be present for it to be manipulated (thinking of Avatar: The Last Airbender imprisonment methods for different benders). As such, you're still creating the elements from nothing.
That being said, there's plenty of flip-flopping through the disciplines as to whether something duplicates the effect of a spell being cast or whether the monk is actually casting a spell.