In Order of the Phoenix there are multiple discussions about travel methods being watched.
“A month?” said Ron, as though he had never heard of a journey lasting such a ridiculously long time. “But — why couldn’t you just grab a Portkey or something?”
There was an odd expression in Hagrid’s unobscured eye as he squinted at Ron; it was almost pitying.
“We’re bein’ watched, Ron,” he said gruffly.
“What d’you mean?”
“Yeh don’ understand,” said Hagrid. “The Ministry’s keepin’ an eye on Dumbledore an’ anyone they reckon’s in league with him, an’ —”
“How’re we going?” asked Fred, looking shaken. “Floo powder?”
“No,” said Dumbledore, “Floo powder is not safe at the moment, the Network is being watched. You will be taking a Portkey.” He indicated the old kettle lying innocently on his desk. “We are just waiting for Phineas Nigellus to report back... I wish to be sure that the coast is clear before sending you —”
If Hagrid couldn't take a portkey because they were being watched, why were Harry and the Weasleys able to take a portkey? Dumbledore was still clearly worried about being watched, as that is the very reason he gives for why they can't use the Floo Network.
Perhaps relevant is another discussion earlier in the book where potential travel via portkey is discussed:
“How’re we getting — wherever we’re going?” Harry asked.
“Brooms,” said Lupin. “Only way. You’re too young to Apparate, they’ll be watching the Floo Network, and it’s more than our life’s worth to set up an unauthorized Portkey.”