Gamma or Beta, but likely Beta.
In Chapter 4, Lenina expresses her disgust for the Gammas as she passes by Gammas boarding monorails after work:
"My word," said Lenina, "I'm glad I'm not a Gamma."
Later, when talking with Henry, it's clear that she is not an Epsilon - and likely an Alpha or a Beta:
"I'm glad I'm not an Epsilon," said Lenina, with conviction.
"And if you were an Epsilon," said Henry, "your conditioning would have made you no less thankful that you weren't a Beta or an Alpha."
Lenina's character throughout the book is consistent with that of someone from the upper classes. She expresses clear disdain at the lower classes, while still docilely going along with the way the World State wants things to go (remember, she uses soma). While many Alphas fit that profile, Lenina's less-imaginative attitude seems to fit a Beta more than an Alpha. It's true that all the major characters in Brave New World are Alphas and Betas, and so we don't know what Gammas, Deltas or Epsilons really think, we should still be able to extrapolate.
It seems like Lenina is a Beta. However, when talking with Fanny while getting dressed (Chapter 3), Lenina is quite clearly wearing green - the color of Gammas! The following excerpt from the conversation between the two is interspersed with a history lesson from Mustapha Mond and the sound of sleep-teaching. All emphasis is mine.
"Do I look all right?" Lenina asked. Her jacket was made of bottle green acetate cloth with green viscose fur; at the cuffs and collar.
"Eight hundred Simple Lifers were mowed down by machine guns at Golders Green."
"Ending is better than mending, ending is better than mending."
Green corduroy shorts and white viscose-woollen stockings turned down below the knee.
"Then came the famous British Museum Massacre. Two thousand culture fans gassed with dichlorethyl sulphide."
A green-and-white jockey cap shaded Lenina's eyes; her shoes were bright green and highly polished.
My guess is that this an oversight on Huxley's part (though the outfit is mentioned once later). I think it's also possible that Lenina decided to dress a bit nicer - after all, she's going out with Henry Foster (to Fanny's dismay). She may not even have been wearing the same green as the Gammas; her jacket is "bottle green" while their clothes are "leaf-green".
Finally, let's take Lenina's job into account. She works as a technician at the Hatchery - a position that needs some intelligence and skill, but is, for the most part, repetitive and monotonous. An Alpha would not be needed to do such work, but a Delta or Epsilon could not do it, and even a Gamma might not be a good choice. The skill needed seems to point to Lenina being a Beta. Also, as Gallifreyan pointed out, Linda is also a Beta, and presumably had a job similar to Lenina's.
In short, Lenina's comments, personality and job all point to her being a Beta of some sort.