In an attack with a single nuke, the BART tunnels seem like a pretty good place to go.
In an attack by Russia or China (or Russia and China), it's not -- unless the people in the tunnels are low in number and have some way to prevent additional people from entering.
In normal times, the tunnels are "actively ventilated", meaning fans are used to bring air in from the outside, but an attack by Russia or China would surely prevent electricity from getting to the fans (because destroying the US's generating and electrical-distribution capacity would be one of the goals of the attack) so CO2 and heat would build up in the tunnels if there are people in the tunnels. Of the 2 dangers, I'm more afraid of the heat (though if the tunnel were in Fairbanks, AK, I'd be more afraid of CO2): the ground is a very poor conductor of heat. Either danger seems likely to kill most of the people in the tunnels before it becomes safe enough to leave the tunnels (2 or 3 weeks).
I think that the subway systems of Moscow and St Petersburg have ventilation systems designed to keep operating after a nuclear attack (because the Russian government takes nuclear war more seriously than governments in the US do) although I welcome more information on that point: I'm particularly interested in details such as whether the air intakes for those 2 subway systems point down (which is the easiest way to prevent fallout from falling into the air intakes and thereby landing close enough to sheltering people to cause their deaths).
I spent an hour or 2 recently Googling for any signs that BART's ventilation system has been designed to keep operating if the electrical grid goes down and I didn't find anything. I know from having seen some of them that BART's air intakes weren't designed to prevent fallout from falling down into them. (I am focusing on air intakes because designing them right strike me as a particularly cheap intervention relative to how much grief they can prevent in case of an attack.)
The tunnels through the tops of the Berkeley and Oakland hills (for roads and maybe train tracks) are much more survivable on my models than the BART tunnels because there is some wind or breeze most of the time in those hills. In general, even when there is no breeze in the flatlands, there is usually a breeze at elevation in the Bay Area.
If I saw credible signs that a major nuclear attack was coming or if an attack had occurred leaving me uninjured enough to walk, I'd head to the Cal Park Tunnel (in Marin County, about a mile from my home) with as much water and food as I can carry. (I'd carry about 93% of the weight as water, 7% as food.) It is 1100 feet long according to some web site (which seems about right to me). I've been in the tunnel often; I notice that even when the breeze outside is barely perceptible, there's definitely plenty movement of air inside (in a constant direction at a constant velocity, which of course is what you want for ventilation).
Now some of you are asking: won't the breeze or wind blow the fallout into the tunnel? Good question! Most of the fallout mass is in the form of "aggregations" between the size of a pea and a grain of sand. Basically what it does is fall out of the sky, then remain on whatever horizontal surface it lands on. It's not that the fallout that is in the form of particles small enough to remain suspended in the air for a while is completely harmless -- if during the hour or 3 during which fallout is falling out of the sky, you can seal off the tunnel or whatever other shelter you are in, do so; failing that, if you have a mask, wear it -- but if I recall correctly my reading of the book Nuclear War Survival Skills, the majority of the danger from the radioactive material that remains suspended in the air or is blown around by winds comes from some process in which beta radiation affects the mucous membranes, particularly in the nose, and is only a significant danger in dry weather whereas of course the air in the Bay Area is moist most of the time.
In short, I think the answer is yes.
The general advice I've seen is to get behind concrete barriers and lay flat on the ground. If you're at a survivable distance from ground zero then the main immediate concern is the shockwave, not the explosion and heat itself. You want to be as low to the ground as possible and ideally below ground so that the shockwave passes over you. Beware of buildings and other structures that might collapse on you, though.
If you're closer to the center I think the chance of survival is low because even if you survive the immediate blast you'll have to contend with the superheated air that will set things on fire, like your clothes and skin. Going down into BART seems a reasonable strategy, though you may not be able to find a part of BART that's deep enough underground and protected enough from what's happening above to survive while in there.
After that the main concern will be fires, radiation, and supply shortages. If you don't already, you'll want to have potassium iodide pills to protect you from the worst effects of the radiation and some basic survival gear.
Unfortunately the Bay Area doesn't really have any fallout shelters or bunkers, so BART tunnels are likely your safest option (as long as trains aren't running!).