For Galveston I doubt you would be able to because of security. Coming over from Bolivar this weekend I was looking to see if anybody was bank fishing there and didn't see anybody. But I did see folks in front of the Army Corp of Engineers building and do everytime I cross the ferry. I also see people fishing on both sides of the ferry landing on the Galveston side. Saw a guy pull in a nice flounder by the picnic tables in that little corner on the right when you get on going to Bolivar.
I see people fishing off the rocks in front of the CG station all the time. It looks like they might have walked over from East beach. It would be difficult to fish that corner on a strong 2 tide day as the water rips around that point. I'm thinking a 4 tide day would be best to fish off of the rocks. Warning...do not wade on the sandbar that comes off of the rocks on the channel side near where East beach ends. People drown there every year due to quick drop offs & strong currents.