These days access depends on keeping the property owner happy. Caves on private property usually have a designated contact and all trips are hopefully scheduled through that person, with the owner's permission. So yes -- joining an active grotto is the way to go. This link might help:
http://www.utexas.edu/tmm/sponsored_sites/tss/links/txgrottos.htm
Beck's Ranch is normally gated and locked and access is through Tx Cave Conservancy, based in Cedar Pk. In fact they are talking about closing it down pretty much all together because of the bats and because of damage that's been done in the past.
Whirlpool is accessed through the Bexar Grotto. You can check their website for trips.
http://www.caves.org/grotto/bexargrotto/ I don't know if it's physically gated but the permission should be obtained before going out. It's best to go on one of the scheduled trip.
Honey Creek is definitely with permission only and only to cavers. Also through Bexar Grotto. It's good to keep in mind that even the water caves are managed by dry cavers. Not really a tech orientation. H.C. is a very long haul of gear through water passages to reach the 1st sump. They already have designated cave divers but they will usually accept volunteer sherpas.
Deep and Punkin are owned and managed by Texas Cave Management. Access by permission only.
Airman's might still be public. I haven't kept up with it, but the rumor is that it's been gated as well.
UT Grotto says it very well:
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WHERE CAN I FIND CAVES TO EXPLORE IN THE AUSTIN AREA?
Most caves in Texas are located on private land. You must have permission of the owner to visit their caves. Trespassing by kids and non-cavers has caused many owners to close their caves to everyone. That is a great loss. Caves are a precious natural resource. They are often very fragile--both physically and environmentally. Some caves have locked gates placed over their entrances, but many of them can still be explored if you get permission from the owner or land manager.
There are a few caves around Austin which can be explored without permission. These are real caves and require some physical ability, helmets, and 3 light sources. There are no trails, no electric lights, and a lot of mud. Contact grotto officers for directions.
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And normally cavers don't post even this much information, but I'm trying to do some damage control. Fireproof UGs on now.