Delta, Gamma, and Epsilon are eliminated fairly explicitly.
On one of her outings with Henry:
"My word," said Lenina, "I'm glad I'm not a Gamma."
Similarly, in conversation with Henry:
"I'm glad I'm not an Epsilon," said Lenina, with conviction.
And again with Henry:
Lenina, meanwhile, had turned her eyes away and was looking perpendicularly downwards at the monorail station. "Fine," she agreed.
"But queer that Alphas and Betas won't make any more plants grow
than those nasty little Gammas and Deltas and Epsilons down there."
We are then left with the possibility of her being an Alpha or a Beta. While there are no explicit statements to adjudicate this, I think there are various indications in the book that point more towards her being a Beta.
Lenina doesn't seem to be as smart as the Alphas. For instance, in this discussion with Henry:
"Why do the smoke-stacks have those things like balconies around
them?" enquired Lenina.
"Phosphorus recovery," explained Henry telegraphically. "On their way
up the chimney the gases go through four separate treatments. P2O5
used to go right out of circulation every time they cremated some one.
Now they recover over ninety-eight per cent of it. More than a kilo and
a half per adult corpse. Which makes the best part of four hundred
tons of phosphorus every year from England alone." Henry spoke with
a happy pride, rejoicing whole-heartedly in the achievement, as
though it had been his own.
Here she seems to ask the question very simplistically, while Henry, an Alpha, gives a precise and technical answer (and it is not even clear that she really understands it).
In fact, there is a particular example of something that she doesn't really understand that stands out as being non-alpha. In a conversation with Bernard, we find the following:
"Adults intellectually and during working hours," he went on. "Infants
where feeling and desire are concerned."
"Our Ford loved infants."
Ignoring the interruption. "It suddenly struck me the other day," continued Bernard, "that it might be possible to be an adult all the time."
"I don't understand." Lenina's tone was firm.
"I know you don't. And that's why we went to bed together yester-
day-like infants-instead of being adults and waiting."
"But it was fun," Lenina insisted. "Wasn't it?"
"Oh, the greatest fun," he answered, but in a voice so mournful, with
an expression so profoundly miserable, that Lenina felt all her triumph
suddenly evaporate. Perhaps he had found her too plump, after all.
"I told you so," was all that Fanny said, when Lenina came and made
her confidences. "It's the alcohol they put in his surrogate."
"All the same," Lenina insisted. "I do like him. He has such awfully nice
hands. And the way he moves his shoulders-that's very attractive."
She sighed. "But I wish he weren't so odd."
The idea of not being infantile at all times is specifically said to be something that Alphas are not conditioned against:
"And I should like to take this opportunity, Mr. Marx," he went on, "of saying that I'm not at all pleased with the reports I receive of your behaviour
outside working hours. You may say that this is not my business. But it
is. I have the good name of the Centre to think of. My workers must
be above suspicion, particularly those of the highest castes. Alphas are
so conditioned that they do not have to be infantile in their emotional
behaviour. But that is all the more reason for their making a special effort to conform. It is their duty to be infantile, even against their inclination. And so, Mr. Marx, I give you fair warning." The Director's voice vibrated with an indignation that had now become wholly righteous and impersonal — was the expression of the disapproval of Society itself. "If ever I hear again of any lapse from a proper standard of infantile decorum, I shall ask for your transference to a Sub-Centre — preferably to Iceland. Good morning." And swivelling round in his chair, he picked up his pen and began to write.
Thus, Lenina's inability to grasp this concept could point to her being a Beta rather than an Alpha.
This one may be a bit of a stretch, but when Bernard speaks to the Director prior to taking Lenina to New Mexico, we find the following:
"I had the same idea as you," the Director was saying. "Wanted to have a look at the savages. Got a permit for New Mexico and went there for my summer holiday. With the girl I was having at the moment. She was a Beta-Minus, and I think" (he shut his eyes), "I think she had yellow hair. Anyhow she was pneumatic, particularly pneumatic; I remember that.
Give that Lenina is described several times as pneumatic as well, it almost seems as if the reader is meant to equate the Director taking Linda to New Mexico with Bernard taking Lenina, in which case it would be reasonable for Lenina to be a Beta like Linda.
Another indication of Lenina being a Beta comes from the following, on one of her outings with Henry:
"What a hideous colour khaki is," remarked Lenina, voicing the hypnopaedic prejudices of her caste.
It is interesting that it says "her caste" in singular, rather than "the upper castes" (as Alphas and Betas are jointly referred to several times). This may indicate that this hypnopaedic prejudice was specific (or more pronounced in) one of the upper castes over the other. And this hypnopaedic prejudice is explicitly shown to be part of Beta conditioning:
There was a pause; then the voice began again.
"Alpha children wear grey They work much harder than we do, be-
cause they're so frightfully clever. I'm really awfuly glad I'm a Beta,
because I don't work so hard. And then we are much better than the
Gammas and Deltas. Gammas are stupid. They all wear green, and
Delta children wear khaki. Oh no, I don't want to play with Delta children. And Epsilons are still worse. They're too stupid to be able..."
In a conversation with Henry about her caste, we have the following exchange:
"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."
If Lenina was an Alpha, it would seem unnecessary to say “a Beta or an Alpha”. After all, he didn’t say “or a Gamma or a Delta”. If, however, she is in fact a Beta then it makes more sense to specifically mention it.