We have been asking for "functioning wi-fi" on BART and CalTrain forever but it looks like it might be happening soon.
BART is in negotiations with a vendor to install Wi-Fi at all 46 stations and on board all of their cars. According to SFgate and BART they hope to have a deal by summer.
The information officer at BART also said that all of the new BART cars will already come ready for Wi-Fi. Sooooo it looks li8ke the change is happening but with most Bay Area transit projects it's going to take time. BART expects to have a full fleet of the new cars...by the end of 2022.