Hello there. I was doing a quick 1v1 with another person on DnDBeyond. For the most part, it was fun. But when I said that they couldn't use mold earth to knock my character prone, they blew up at me and said that I was boring and that I was limiting how the game should be played. The reason I said that they couldn't use the spell in this way is because it's not mentioned anywhere in the spell, and because it would be easily abused as a cantrip. My viewpoint is probably biased, but I want to know if I did anything wrong, and if so, what I should do differently in the future.

I think you're ok, but I think it's a bit of a wobbler, but more leaning toward your interpretation.  While the actual "earth moving" property does beg the question of what would happen to a person or object standing on top of that moved earth, the only explicit tactical effect the cantrip is explicitly listed as capable of of producing is turning the ground into difficult terrain.  It's also stated the force used in the earth moving iteration is not strong enough to cause damage, and I'd infer also the momentum to knock someone over.  At best, maybe the cantrip can force a reaction to be made to keep your footing (negating attacks of opportunity or other reactions), but as I think through it I'm thinking you're more in the right ... though I can see other DM's "rewarding" the player for creativity and not realize the game precedent they set with The Cantrip Tripmaster.

If you were in pursuit of the caster and using the DMG's chase mechanics, I guess you could treat it as a slip check ... but the caster has already made the ground difficulty terrain so that seems to be effect stacking on a cantrip.  Of course they could also create a five feet drop which maybe you'd have to make a DEX check to avoid falling in as opposed to negotiating around like difficult terrain.

I'm sorry you caught fire for your decision, though I imagine things might not have been going well if the contest came down to mold earth being the bellringer.

You're a GM. You made a ruling - that's part of your job: you adjudicate the rules.

From your statement, you don't sound like you did it frivolously, but actually considered the larger world implications if the cantrip worked the other person wanted it to.

I only have one side of the story, and incomplete information - but it doesn't sound like you did anything wrong to me.

 People just have different approaches to RPGs. Some want rigorous, clearly defined rules ( and thus really like RAW, since that's objective, external "law" as to how the game it to be played ). Some expect that "the rule of cool" is going to take precedence over everything else. This might be what happened here: the other person thought that a "cool" effect should take precedence over a rigid set of rules, or in-world verisimilitude. 

This simply may have been a disconnect in what you both emphasize in the game, and differences in expectations as to how the game should be run.

In the future, maybe getting a better handle on style of play an expectations from the other people in the gaming group might help avoid such events.

You are correct in the way you interpreted what can and can not be done with a cantrip. Mold Earth specifically shows that it's not an attack spell as it doesn't say you can target anyone with it. It also says it can't do any damage. The only thing you can do to use it against someone else is create difficult terrain.

This however is your call, and you made it. Don't let someone get you down because you made a game decision they didn't like.

If the spell were meant to do damage or have an effect like knocking prone, then it would say so. 

Creative spell use is fine but in this case, the PC used the wrong tool. Even if they dug a hole directly under a creature, it would not fall and take damage.

You’re the DM. A player whining about a DM’s ruling is always in the wrong.

Does it make it that imbalanced? Was the player trying to knock you prone at get advantage every round? How would they do so, after already using their action on the spell. I suppose it could be OP with multiple PCs where the druid knocks the enemy prone and the barbarian, fighter, and rogue all wail on him.

One way you could resolve it with fewer complaints is allow it, but have the target make a Dex save. You don’t have to tell the player the DC or the target’s roll, and you could make it very easy, like 5. I don’t think using an action to render an enemy prone would ever be OP if it only works 1/4 of the time. You can achieve just as good with a basic Shove action, although you have to be in melee range. Most spell effects have a save. You can dodge a wall of stone or a fireball, so I think you can dodge a mold earth.

Earth Tremor is a first level spell that can knock a person prone if they fail their save. That is the spell the player wants to be using to get that effect. Mold Earth is not that spell.

What makes sense vs what is written in the description. A lot of things “make sense” but what is written is inarguable, if mold earth knocked targets prone it would say so in the description. The game is in fact, quite literally meant to NOT do the things they were claiming it should. You’re fine.