You have a number of options. The easiest is to just leave the interior unfinished. If you don't want to do that, a good grade of polyurethane or lacquer should be fine. Another option is to just cut some pieces of cardstock and stick them to the sides with double-stick tape.
Avoid latex finishes (note--"water based" and "latex" are not the same anymore--there are waterborne urethanes for example). Shellac would be risky, if it's freshly mixed it should be OK, if it's old it might never dry properly. Don't use a non-polyurethane varnish unless you have tested it and know for sure that it's OK--some remain a bit tacky, some dry hard, some get tacky when it's hot and humid.
For another option, purpleheart will take a high polish with no finish at all and as exotic woods go it's even fairly cheap.
If you use an oil based paint or varnish you should have no problem with cards sticking. You do not need humidity for cards to stick to latex/water based enamel paints.
If you are l9ooking for a clear finish I would recomend any one of the gel wipe on varnishes.
Like drawer interiors, you can leave it unfinished. I find a spray can of lacquer works very well for small projects--fast and easy to apply, cures quickly. A second choice would be polyurethane, one of the toughest fiishes.
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