Moving to Bay Area with two small children - clueless!

Perhaps you could let us know which hospital, or generally what you mean by "downtown SF"? "Downtown" isn't a common SF designation. Several hospitals could be described as central, but they're in different districts with different kinds of transport access, so recommendations could be different. The UCSF system, for instance, offers shuttles between its locations, and that could make a big difference to what's realistic for you.

Albany or El Cerrito would be good, family friendly and close to BART.

What is your budget? SF is insanely expensive and you don't want to live downtown with kids. By downtown, I assume you mean the financial district or somewhere on Market Street. I would suggest that he plan to take BART or an express bus to work and that you live outside the city. But then the tricky part is figuring out which direction and how far you should be from the city. A huge percentage of people on this forum are in the Berkeley/Oakland area, so you will probably get a lot of votes for that area. The commute from there, assuming you live close to a BART line, is quite reasonable, but if your budget can't handle that area, you'll need to look further and further away from the city (we refer to SF as the city). Prices drop in places like Pleasant Hill and Hercules, and drop more if you go out to Antioch and Brentwood, but then the commute goes way up.

If his fellowship is through UCSF, you should look into their family housing. The apartments are located at Mission Bay and Parnassus campuses, and as someone mentioned, there are shuttles across the campuses if he needs to rotate between hospitals. 

Do you want to stay in the city of SF or be in a more suburban area? If you stay in SF finding a preschool with openings now will be hard, although I am sure there are some. If you move out to the East Bay area (Oakland, Berkeley, Albany, El Cerrito, etc) the commute into SF will be about 45min-1hr via BART. Moving out to the North Bay or Peninsula will be a 1 hour + commute since there isn't much public transportation. All areas have good school options (although some better as kids get older) and safe neighborhoods. I would suggest reading about each area on BPN and going from there.  Good luck. 

I’m trying to think of a hospital that’s truly downtown.   Not trying to nitpick the terminology, but “downtown” means a specific area of SF (financial district, Union Square & immediately surrounding area) and exactly where he’ll be working it affects the transit situation a lot.  St Francis, Zuckerberg, UCSF are not downtown, for example, and all would involve secondary transit from BART of differing degrees of difficulty.  Albany and El Cerrito are good ideas if the job is reasonably close to BART.  If not, think through getting from BART to the hospital.

Neighborhoods within easy walking distance to BART will make commuting to SF doable and even sane. We live in North Berkeley and the weekday commute to downtown SF is about 40 mins door to door. There are good preschools, playgrounds (Totland!), tennis courts, swimming pool, markets, restaurants/cafes/coffee shops, museums, theaters all within walking distance in our neighborhood. There's even a farm and great hiking trails within a 10 min drive. The hitch? Finding affordable housing. Welcome to the Bay Area! It's really a wonderful place to raise a family if you can find a suitable home. Best of luck! 

You might want to listen to a podcast called "Bay Curious".  https://www.npr.org/podcasts/500557090/bay-curious  Specifically the episode "Where have all the Kids in San Francisco gone?"  Downtown if far from family friendly and San Francisco has become so expensive families with young kids are moving out.   Oakland, (can be okay to really bad) so that leaves Piedmont, Berkeley, Orinda, San Leandro, Richmond, Pacifica, Marin and the question of can you afford it?  Depending on where you live it will take your husband well over an hour just to get to downtown San Francisco by public transit and even longer by car.

Hope this helps; best of luck.

Since most (all) of the responses you have gotten recommend living outside SF, I thought I would put in a vote for living in the city. (Disclosure: We live in Berkeley, within walking distance of a BART station, which is great, but I grew up in SF, in the outer Richmond.) I think you can find something affordable (in Bay Area-adjusted terms) in the western part of the city: Sunset, Richmond, etc. It will ease your husband's commute to not have to cross the Bay to get to work (I think that is also a psychological advantage). There are simply more things to do in SF for families from museums to beaches; the great outdoors (Marin, San Mateo coast, East Bay hills) are just a short drive away. Many of the playgrounds in SF are new and fantastic, while in Berkeley we have hardly any. If you are only planning on living in the Bay Area temporarily, I think you will get more out of your time to live in SF rather than a suburb, which might not be too different from where you are coming from. My two cents.

Re the downtown hospital, which as others reported is confusing as there isn't one, at least that I know, maybe you mean St Francis in Nob Hill - regardless - the thing I wanted to caution you on is that although it may appear only a short walk from "downtown" (ie the bart stops at Union Square or the Financial district) it is up a very big hill so unless your husband likes to walk that, he'll need to transfer from bart to a bus. Which is fine, just keep it in mind that when looking at SF maps, the distances are misleading due to the hills. 

Here are a few SF neighborhoods to consider -

1. Potrero Hill - easy access to downtown areas and you see lots of kids and families walking around- which is unusual in many neighborhoods in SF

2. South Beach - I love this neighbor and it would work great if you have toddlers but not sure once kids become school age.

3. The Marina is also family friendly.

I am sure there are other areas out in the avenues that are more family (the sunset or richmond). As everyone says, these are very expensive.  You could consider Alameda and have him commute by ferry. Alameda is definitely kid friendly and somewhat more affordable than some other places.

Wish you the best of luck!

Bernal heights, Glen Park, Daly City areas are very family friendly. There is a sweet preschool, Room to Grow, in Bernal Heights. UCSF area is also very family friendly. Note that Daly City and Richmond/Sunset neighborhoods are colder and foggier than bernal heights, which has a nice sunnier weather.