Which neighborhood for a job at Stanford

You can find many communities within 15 miles of Stanford that are culturally diverse, one suggestion is Redwood City. Housing prices are quite high however and you may want to rent for a while to be sure you find the right fit. The schools are good but are not known as the super highly ranked schools seen in Palo Alto and some of the communities to the South of Stanford. Another option, with more reasonable housing is East Palo Alto. Many people affiliated with Stanford and Facebook now live there and Facebook has contributed to the schools. Best of luck with your relocation.

The answer to your question is not as straightforward as it might seem. I’m a native East Coaster and lived in the Boston area for many years, and (one) major issue is that the built environments in New England and the Peninsula evolved very differently. Stanford/Palo Alto has a cute downtown area, but most of the peninsula is suburban sprawl of a type that doesn’t really exist in New England. The peninsula evolved from fruit orchards, which were replaced by bedroom communities, as opposed to The Boston area’s growth based on 400 year old towns with small farms surrounding them. This is all relevant because, whereas Cambridge is 15 minutes from Somerville, Watertown, Belmont, Arlington, etc., towns are geographically much bigger in the Bay Area. For example, Cambridge is about 7 sq. miles; Palo Alto is 26. The “town” of Stanford is little more than the campus; it isn’t really a town.

Because of this, the rudimentary nature of public transit on the peninsula, being the tech center of the US (of not the world), and the horrific traffic, 15 minutes away holds little meaning. East Palo Alto is about five miles from Stanford, but it is unlikely you could travel between them in less than 20 minutes except in the middle of the night. Menlo Park (3-1/2 miles) is doable in about 15 minutes. Maybe Atherton (4-1/2 miles), perhaps Woodside (7 miles). 
Many of these communities have large non-white populations, but this doesn’t always translate to diversity. There are large numbers of people from abroad who immigrated or are here on H-1 (work) visas; many came from India and China. There is a significant Latinx population as well. However, class divisions are striking. In general, Latinx populations are lower income and work in the service economy, while Asians are highly educated, skilled tech workers. The Black population in Northern California is tiny compared to that of Northeastern cities. East Palo Alto was a majority-Black city, but it has been a gentrification target and much of the Black population has been displaced. In sum, you will find people from a diversity of places, but the multicultural environment of a place like Cambridge is elusive. Obviously none of this is absolute, but I’m looking at census data, not relying on personal impressions.

Even less common is economic diversity. For example, median household income is $107k in Cambridge, and 12% of Cambridge residents live in poverty. In contrast, Menlo Park’s median household income is $168k, with under 6% in poverty. And Menlo Park isn’t an outlier. Woodside and Atherton’s medians are over $250k; less than 5% and 3% of their respective  populations live in poverty.

There are places more like Cambridge with regard to social and economic diversity, but I doubt you’ll find that within 15 minutes of Stanford. East Palo Alto is probably closest geographically. San Jose might be most “like” what you seek, but it is a huge, sprawling city spread over 181 sq miles. Oakland and Berkeley, both northeast of Stanford, are really the best matches for your criteria, and great places to live, but they are an hour or more drive from Stanford, depending on traffic, and at least 90 minutes by transit.

Best of luck!