In The Fellowship of the Ring
after failed attempt to go through Caradhras the Company discusses the Moria way.
Then Aragorn warns Gandalf of passing through the Moria.

'I will,' said Aragorn heavily. `You followed my lead almost to disaster in the snow, and have said no word of blame. I will follow your lead now - if this last warning does not move you. It is not of the Ring, nor of us others that I am thinking now, but of you, Gandalf. And I say to you: if you pass the doors of Moria, beware! '

What exactly is he implying here? Did he know of Balrog or what exactly is of so much danger to Gandalf but not to the others?

Aragorn knew of the dangers of Moria

Moria was plain notorious. The name "Moria" itself means The Black Pit in Sindarin. The fall of Khazad-dûm was well-known in Middle-earth, and it became widespread knowledge that there was a nameless terror, or to the Dwarves, Durin's Bane, lurking in the former Dwarven kingdom. So much so that even the Hobbits had heard of the name Moria.

'There is a way that we may attempt,' said Gandalf. `I thought from the beginning, when first I considered this journey, that we should try it. But it is not a pleasant way, and I have not spoken of it to the Company before. Aragorn was against it, until the pass over the mountains had at least been tried.'

'If it is a worse road than the Redhorn Gate, then it must be evil indeed,' said Merry. `But you had better tell us about it, and let us know the worst at once.'

''The road that I speak of leads to the Mines of Moria,' said Gandalf. Only Gimli lifted up his head; a smouldering fire was in his eyes. On all the others a dread fell at the mention of that name. Even to the hobbits it was a legend of vague fear.

_{The Lord of the Rings: The Fellowship of the Ring}

Aragorn had spent most of his life in the Wilderness, and his childhood as a fosterchild in Rivendell. It isn't unreasonable at all to say that he had heard of the legend of Moria. Furthermore, he himself had passed through Moria before. The answer here suggests that Aragorn was possibly captured by Orcs in Moria, so Aragorn knew of the danger of Moria and warned Gandalf of it. Alternatively, we could speculate that Aragron felt the presence of the Balrog, but there isn't concrete evidence of that. No one knew that there was a Balrog in Moria, only that it was a nameless terror.

"I too once passed the Dimrill Gate but though I also came out again, the memory is very evil. I do not wish to enter Moria a second time."

_{The Lord of the Rings: The Fellowship of the Ring}

Gandalf himself had no idea of the identity of the nameless terror until he saw it. So Aragorn could not have known either.

What he did know was that there was a danger in Moria; likely a large hive of Orcs and/or something bigger. Realistically, Aragorn could have believed that the only person in the Fellowship able to deal with that something bigger would be Gandalf, a Wizard.

Aragorn's foresight

Aragorn was a descendant of the Númenóreans, who were reputed to have incredible gifts of foresight. The very quote you cite is very likely to have been one of his foresights. Immediately after escaping from Moria Aragorn says this:

"Farewell, Gandalf!" he cried. "Did I not say to you: if you pass the doors of Moria, beware? Alas that I spoke true! What hope have we without you?"

_{The Lord of the Rings: The Fellowship of the Ring}

And a much more explicit use of the word (Appendix A):

'"I see," said Aragorn, "that I have turned my eyes to a treasure no less dear than the treasure of Thingol that Beren once desired. Such is my fate." Then suddenly the foresight of his kindred came to him, and he said: "But lo! Master Elrond, the years of your abiding run short at last, and the choice must soon be laid on your children, to part either with you or with Middle-earth."

_{The Lord of the Rings: The Return of the King}

There is evidence that other descendants of the Númenóreans had the gift of foresight. Denethor, for one, and his fellow Dúnedain Halbarad. He says this right before entering the Paths of the Dead.

This is an evil door and my death lies beyond it. I will dare to pass it nonetheless; [...].

_{The Lord of the Rings: The Return of the King}

His foresight is proved correct, when he dies while carrying Aragorn's standard into the Battle of the Pelennor Fields.

So to answer your question, I believe it was Aragorn's pre- and firsthand knowledge of Moria, together with the Dúnedain's powers of foresight, that prompted him to say what he said.