I'm afraid there are no perfect spots for you but I'll share what I can. BTW, I'm from SF but have lived in Nevada for about 40 years. Love to visit SF.
The only place we have stayed with the trailer that is close to SF is SF RV Resort
in Pacifica, just south of SF. I know you said no resorts but this is actually more of a parking lot. The best thing about it is it's proximity to SF and that it sits on a bluff along the ocean. The sites, if you can call them that, are very narrow and cost about $50. No handy bus service to the city but once there, the local buses are inexpensive.
Half Moon Bay SP
might be an option for you. It's also $50, located about 25 miles south. No hook-ups but hot showers are available and a dump station. It's less than 10 miles to I-280 which would be a quick drive into SF.
The only RV park within the city limits is Candlestick RV Park
, right across the street from the football stadium for about $75. It looks like a parking lot but considers itself a 5 star RV Park. I don't like to recommend this one because of it's location. But, judging from the website, it looks secure enough, if you don't wander off the grounds at night. The best thing about it is a shuttle bus to the downtown area, $12 round trip.