The article from which this claim stems has multiple issues. The most glaring is that even though its subtitle is "The 30 Million Word Gap by Age 3", the body of the article never actually makes this claim, instead claiming that there is a 32 million gap by age 4.
The authors of the article gathered data on how many words per hour a child heard through monthly 1-hour observations starting at 7-9 months of age and ending at 33-36 months of age. How and when this observation was conducted is not described. On average during this observation, children in families on welfare heard an average of 616 words while children from professional families heard an average of 2153 words. They assume that children are exposed to the same number of words per hour for 14 hours/day, and conclude that:
In four years of such experience, an
average child in a professional family would have accumulated experience with almost
45 million words, an average child in a working-class family would have accumulated
experience with 26 million words, and an average child in a welfare family would have
accumulated experience with 13 million words.
This of course doesn't match the claim in the subtitle (since it refers to age 4). A couple of severe problems in the analysis stand out:
The assumption that children are exposed to the same number of words for 14 hours per day as in the one hour of observation is suspect, and is impossible to evaluate since the conditions of observation are not described.
Presumably people talk less to babies than to children who can understand them, so extrapolating the constant rate of exposure back to birth is unreasonable.
The math is also a little off: 2153 words/hour for 14 hours a day for 4 years works out to 44.01 million, not 45 million. Based on their numbers and assumptions, the actual word gap by age 3 is 23.6 million words.
The article also seems to have never been subject to peer review, appearing first in a book by the authors and later being reprinted in the magazine "American Educator", published by the American Federation of Teachers. Google Scholar lists the correct citation as
Hart, Betty, and Todd R. Risley. "The early catastrophe." Education Review 17.1 (2003).
However, the only journals I was able to find with that name are a journal of book reviews and what appears to be a magazine, neither of which published it.
On the whole, it seems that the word gap is large, but the claim probably significantly exaggerates it.