It's said that Dementors are blind. If someone escaped from Azkaban and the Dementors went looking for them, how would they know if they find the person they're looking for?
They still have ways of recognizing people.
When Barty Crouch escaped, he had his mother using Polyjuice Potion covering for him.
“The Dementors are blind. They sensed one healthy, one dying person entering Azkaban. They sensed one healthy, one dying person leaving it. My father smuggled me out, disguised as my mother, in case any prisoners were watching through their doors. ‘My mother died a short while afterwards in Azkaban. She was careful to drink Polyjuice Potion until the end. She was buried under my name, and bearing my appearance. Everyone believed her to be me.”
They do have ways of “seeing”, despite their blindness.
It’s also the job of the Ministry, more than the Dementors.
The Dementors might not be able to track down and recognize a specific escaped prisoner - however, it doesn’t matter because the Dementors aren’t solely or mainly in charge of finding and returning escaped prisoners to Azkaban. When Sirius Black escaped, it was the Ministry of Magic who were in charge of recapturing him.
“They still haven’t caught him, then?’ he asked.
‘No,’ said Mr Weasley, looking extremely grave. ‘They’ve pulled us all off our regular jobs at the Ministry to try and find him, but no luck so far.”