In chapter 23 of Half-Blood Prince, Dumbledore answers this (emphasis mine):
"The snake [Nagini - obviously]?" said Harry, startled. "You can use animals as Horcruxes?"
"Well it is inadvisable to do so," said Dumbledore, "because to confide a part of your soul to something that can think and move for itself is obviously a very risky business. However, if my calculations are correct, Voldemort was still at least one Horcrux short of his goal of six when he entered your parents' house with the intention of killing you. [...] I am sure that he was intending to make his final Horcrux with your death. As we know, he failed. After an interval of some years, however, he used Nagini to kill an old Muggle man, and it might then have occurred to him to turn her into his last Horcrux. She underlines the Slytherin connection, which enhances Lord Voldemorts mystique; I think he is perhaps as fond of her as he can be of anything; he certainly likes to keep her close, and he seems to have an unusual amount of control over her, even for a Parselmouth."
It makes sense to not use a living thing as a Horcrux: as we've seen with Harry (and also Nagini), all it takes to destroy one is to kill it, and something that can walk around on its own could accidentally kill itself in any number of ways.
Voldemort only resorts to this when he has very few other options. Nagini, for example, is the last Horcrux he creates, as confirmed in a 2007 live chat on the Bloomsbury website:
Lady Bella: Whose murders did voldemor [sic] use to create each of the horcruxes
J.K. Rowling: The diary - Moaning Myrtle. The cup - Hepzibah Smith, the previous owner. The locket - a Muggle tramp. Nagini - Bertha Jorkins (Voldemort could use a wand once he regained a rudimentary body, as long as the victim was subdued).
J.K. Rowling: The diadem - an Albanian peasant. The ring - Tom Riddle snr.
We can speculate, as Dumbledore does in the HBP quote above, that Nagini was turned into a Horcrux because Voldemort, recently returned to a physical body (albeit a weak one), desperately wanted to complete his "seven Horcrux" spell. Nagini was a relatively safe choice for a "vessel" because of Voldemort's level of control over her and because, especially at this stage of Voldemort's life, she's never far away.
At any other time in his life, it would have been too risky.