Water-Safe Crystals:
Clear Quartz
Rose Quartz
Amethyst
Smokey Quartz
Citrine
Agate
Moonstone
Carnelian (though NOT safe in salt water)
Rutilated Quartz
Jasper
Tiger's Eye
Aventurine
Black Obsidian - (FYI: can break in EXTREME cold or hot water temperatures)
The crystals listed above are a great place to start and are especially safe when in polished, or tumbled form. Always double-check when it comes to salt water because salt can be corrosive!
Water Unsafe Crystals:
Fluorite (especially SALT water)
Selenite
Apophyllite
Tangerine Quartz
Gypsum
Lepidolite
Opal (although Australian Boulder Opal is generally safe because it is not porous)
Azurite
Apatite
Halite
Malachite
Turquoise
Ulexite
Labradorite (I've safely placed Labradorite in water in short bursts and it was fine, but do not count on my experience as the norm)
Celestite
Pyrite
Hematite
Many of the crystals on the unsafe list will completely dissolve when placed in water, or they will lose their shine. Either way, play it safe when it comes to these minerals and ALWAYS DYOR (do your own research)