In CoS, Fudge comes to Hogwarts to arrest Hagrid, so he can make it look like the Ministry is doing something (i.e., arresting innocent people). But why did Fudge himself have to come? Isn't this a waste of time? Couldn't he just send a few Dementors, or some lowly wizard in the Ministry?
I believe that Fudge came due to both reasons mentioned in the other answers:
- He was pressurized (or bribed) by Lucius Malfoy to ensure Hagrid's arrest.
- Fudge was there to use his authority as Minister for Magic to ensure Dumbledore did not cause any trouble and this arrest would go as smoothly as possible.
Fudge does have a history of popping up as though he has not much work to do.
- He was there at the Leaky Cauldron when Harry got off the Knight Bus after attacking his Aunt Marge in PoA.
- He was readily available to replace Percy Weasley (who replaced Bartemius Crouch Sr.) as a judge in the Triwizard Tournament in GoF. Considering Percy was one in the second task, the judge clearly did not have to be someone in such a high position as Minister for Magic.
- He came along to arrest Dumbledore as well in OotP (though Dumbledore, being Dumbledore, did earn the right to be arrested by the Minister for Magic himself).
But all this does seem to beg the speculation that Fudge had a lot of free time on his hands. I highly doubt Prime Ministers and Presidents of countries would have such free time in their schedule to take care of such trivial matters personally. Maybe the Ministry of Magic runs pretty smoothly with the Minister not really having a lot on his plate. Or maybe Fudge uses Time-turners to take care of all his official duties apart from stuff like this.
Fudge knew that Dumbledore wasn't going to agree with Hagrid's arrest. If he had just sent a few Dementors or some lowly wizard, they could have been chased away by Dumbledore.
He could tell them that Hagrid was on vacation, or something.
By coming himself, Fudge brought the whole Ministry's power to ensure that he will have what he wants.
One could argue it was a narrative device to show how flawed Fudge was as a character, as the only reason he was there was because Lucius Malfoy wanted Hagrid, who he deemed inferior due to his 'half bred' breeding, to be arrested. Lucius was aware that Dumbledore would resist which is why he brought Fudge, which undermines the validity of the Ministry of Magic whilst support Lucius' character as one that lusts power and intimidation.