The bit quoted in the question tells one important thing up front:
1. Riddle has the means and ability to successfully tell the Basilisk to target a specific person.
We know a Basilisk can follow orders (from both Magical Beasts and from CoS: "It won’t come until it is called,” said Riddle calmly).
When Riddle says his target has been Harry for months, he doesn't have very many means other than the Basilisk with which to do so - Ginny couldn't be seen behaving too suspiciously out in the open; and Harry ends up hearing the Basilisk in the pipes many times even outside of Myrtle's Bathroom. So it doesn't seem too far-fetched to assume that statement is implying Tom has been having Harry followed around by the Basilisk. Reason being that he knows Harry won't be able to help 'playing the hero' and thereby increases their chances of 'meeting'.
This would also explain, beyond the huge coincidence it looks like, in the books, how Harry is the first one to literally stumble into Justin.
If Tom can tell the Basilisk to specifically target one person aka Harry, presumably he can tell it to do the same for others.
It also tells us that targeting Harry has replaced killing Mudbloods, so it doesn't seem too much of a stretch to assume that the method being used for both was the same; aka specific targeting - the Basilisk cannot be using some 'intrinsic' quality of Muggleborns to target them, because it could not have used that quality to target Harry.
But there's another thing that points to this.
2. Every single Muggleborn attacked was one who would be known to Ginny, and who Ginny would have had reason to talk to Tom about.
We know that Ginny had been confiding to Tom about her school life, and all about her crush on Harry, and all about Harry's school life too.
Colin Creevey: Ginny's Year & Housemate; Both had massive fancrushes on Harry was noted by Ron; Sat next to Ginny in Charms.
Ginny Weasley, who sat next to Colin Creevey in Charms, was distraught...
Justin Finch-Fletchley: The boy who accused Harry of being Slytherin's Heir; the one who was highly instrumental in the drama that led to the reveal about Harry being a Parseltongue - an event Ginny would certainly have talked to Tom about!
Hermione Grander: Close friend of Harry and Ron; Cleverest Witch in her year. Presumably Ginny would have talked her both as a friend of her brother, friend of her crush, and for her own "Hermiony-ish"-ness, even if the two of them were not yet friends.
Penelope Clearwater: It is possible she was petrified only because she was with Hermione at the time, but even she is someone Ginny would have had cause to talk to Tom about - she was Percy's "secret girlfriend", and Ginny was the only one aware of this.
Not one of people attacked is a random Muggleborn who Ginny would not have had reason to know, or reason to talk about. Which lends even more credence to the idea that the target needs to be specifically pointed out to the Basilisk.
3. As for how the target is recognized by the Basilisk...
I don't think we have any concrete idea of that. We can speculate, though. Some speculation:
It couldn't be by face, because being hidden in then pipes, the Basilisk would hardly be able to see someone from within without also exposing itself - at which point whether it's seen by the 'right' person or not becomes a moot point. So the basilisk would need to have a way to recognize target before it confronted them.
Tom was having Ginny pick up some item belonging to the targets, which was then used for tracking;
The basilisk might be capable of understanding human speech and identifying students from their conversations and tracking them (by footsteps? smell? body heat? heartbeat?) to situations where they were alone. This could explain why the basilisk attacked Justin in the presence of Nearly-Headless Nick - the basilisk just realize Nick was there.