How did Mr.Poe know that the kids were at Briny Beach?

[Dante]

The Baudelaire children are stated to be spending the day at Briny Beach, so it's reasonable to suppose that many hours may have passed since the destruction of the Baudelaire mansion. That allows plenty of time for it to be established from neighbours and other witnesses that the children were not at home at the time, and for people who know the family and the children - as indeed Mr. Poe did - to suggest where they might be found. Mr. Poe's duties as executor of the Baudelaire estate are a function of his job as a banker and as such it's quite likely he does have superiors who provide some direction to his actions; but we might also conceivably read that use of "sent" as more indicating that he was told where to find the children as that anyone told him he would have to.

[Optimism is my Phil-osophy]

After years of reading, I came to the conclusion that Beatrice (or Beatrice and Bertrand) asked the banker to fetch their children from the beach, as the fire was planned. The banker was to leave the children with their next of kin during the weeks when Beatrice (or Beatrice and Bertrand) remained hidden inside the tunnel. Something similar happened just before the Baudelaires went to the Hotel D. J.S. asked the banker to pick up the Baudelaires who had just arrived at the beach. This is a strong indication that J.S. was Beatrice disguised as Jacques Snicket.After all, Beatrice never wanted her children to get deeply involved with VFD, despite teaching them at home the skills of a VFD agent.

[Foxy]

Perhaps he had been searching all over the city for them, and finally found them at the beach? You bring up a good point, how did he know where to go? Dante is probably right, someone pointed him in the right direction. The scary thing is, the arsonist may have been the one to tell Poe where to go, because perhaps that person was spying on the family. The clues in TBBRE say the parents sent the children to the beach.