If you are charging them, using an ordinary charger, then they should be vented, because a higher charging rate will cause the electrolyte to "boil" and this produces gas, which is flammable. If you are using a battery maintainer such as the Battery Minder I use or the Battery Tender you can find in Home Depot and elsewhere, the charging current is so low there is no gassing.
Solidly mounted? Only if you have it on a moveable platform that could cause them to tip over, which would cause the electrolyte to leak out. It's actually pretty hard to tip a battery over as you know. Just don't put them in a spot where you could run into them with your car or possibly damage them in a similar manner.
I'd cover the posts with the plastic caps that come on new batteries to prevent an accidental short if a wrench falls on them or something similar. That could start a fire! If you don't have the caps, I'd cover the posts with electrical tape.
My batteries stay in the trailer and are hooked up to my Battery Minder.