We are not comfortable living in white enclaves however, my husband and daughters are not white and I do not want my daughters to feel like they are different for the color their skin. It is one of the reasons we are moving back to a big city.
I feel similarly, but that is a big part of the reason we live in the city and use public schools. Roland Park, Mt Washington, Medfield and Garrett Heights are majority-minority schools. Even Hampden is only 62% white, 38% minority. The only private schools in the area that I know about that approach that level of racial diversity are Catholic, which is a non-starter for my family. Park, which is the 'progressive' private school in the area, is 68% white. Most of the other privates with good academics are significantly whiter than that - Friends is 75%, etc. Wilkes is an exception, but it's located downtown, so not going to be convenient from Towson.
As for white enclaves, those exist for sure - including in Towson. Even if you don't plan on using the public schools, I'd check the demographics of the local zoned school to get a sense of the racial mix of the various neighborhoods. Towson is whiter than the city, by a big margin, although perhaps less starkly segregated. The specifics matter here; living in Baltimore as a black person feels different from living here as an Asian person, because the racial history of the area has been deeply structured by specific forms of anti-black racism.
FWIW, we live in one of the 'white enclaves' mentioned above and have both black and Asian families on our block; parts of of these neighborhoods have gone through rapid demographic transformations recently. It's also true that the neighborhoods around Hopkins and Loyola have lots of faculty and similar types; most of the white people are the 'do-gooder liberal' sort (which does not, I realize, preclude racism - but it's the kind of place where lots of white people have Black Lives Matter signs in their yards, to give a sense of the culture). This map of voting in the 2016 election gives a reasonably good political lay of the land:
https://www.nytimes.com/interactive/2018/upshot/election-2016-voting-precinct-maps.html#11.80/39.320/-76.620/35901.
My point here is not that Democrats can't be racists; more that given how Democratic Maryland is overall, the red zones on the map are a pretty accurate representations of places I wouldn't be excited to live as a minority.
On the flipside, to really prioritize racial diversity and a family-friendly neighborhood, my first choices would be Bolton Hill (there are affordable pockets), Reservoir Hill, and Hamilton-Lauraville and surrounding areas in the northeast of the city.
And one more thing on schools - it's true that the flagship schools are oversubscribed and you need to live in the boundaries to get in. It's also true that there are plenty of genuinely bad schools. But you don't need to live in-zone to send your kid to Hampden or Garrett Heights, and probably not Medfield either. Plus lots of other good places that aren't as convenient to Towson (Mt Royal, Cecil, etc).