No More Than Usual
There is no direct evidence to support the fact that Dumbledore sent anyone to watch Harry upon the escape of Sirius Black. Nor, I think, would there have been any need.
At first, Harry is living at Privet Drive, and by everything we can see does not leave the house while on break. Since the house is protected from Voldemort/his supporters by Lily's spell, Sirius can't get at him. Guards might even be a detriment in that situation.
After Harry runs away he's quickly found and set up in the Leaky Cauldron and told to stay in Diagon Ally. He's literally surrounded by adult wizards to include friends/associates of Dumbledore, so no need for Dumbledore to post a specific guard.
Then Harry's at school, and at that point with Dementors guarding the perimeter and teachers around there is again no need for a specific guard.
I'll point out that Mrs. Figg, Harry's squib neighbor, is known to be friends with Dumbledore and based on scenes from Harry Potter and the Order of the Phoenix onward may have been proving Dumbledore updates on Harry's doings. So I would assume she'd be keeping Dumbledore apprised of the goings-on at Privet Drive and if Harry'd taken to wandering around alone perhaps a guard would have been set.