Moving to the Bay Area -- should I bring my crash pad??

I'm moving from Colorado to Palo Alto in September. I prefer steep, overhanging sport climbing (Rifle, Rumney, the Red, etc.) but boulder occasionally when that's all that's available. 

Is there enough good bouldering around the Bay Area to warrant bringing my crash pad? Or is it far enough away that I'd be better off sport climbing instead?

Unless you're really strapped for space, I'd bring the crashpad if you're into bouldering. Castle Rock State Park, Indian and Mortar Rock in Berkeley, Vacaville/Nut Tree, Goat Rocks, etc., etc.. There's much more bouldering than sport climbing within the ~1 hour range. You have to drive more in the 2+ hour range to get to the sort of sport climbing you'd be into.

Yes, plenty of bouldering to be done near Palo Alto. Castle Rock and Mt Tam are pretty close by. Some nude beaches along the coast that are okay too.

I would say keep it unless you are really, really tight for space. You will quickly exhaust all local climbing options (sport, bouldering, and otherwise) and need to travel several hours in a car to get to the destination areas, where there is plenty of world class bouldering.