When I buy hot chilies, I look for firm, uniformly bright (or dark, depending on the type) green specimens. Unless I'm in a big hurry, I pick through the pile and choose them individually. I avoid any that are soft, discolored (including the ends), missing caps (the part that attaches to the stem -- they spoil faster after this is removed), wrinkly or otherwise look damaged or old.
If your store doesn't have any that look bright green, crisp and plump, try to find a new place to buy them, or just buy them in smaller quantities and get the least disreputable specimens and use them immediately.
You want to buy chilies from a place that has a high turnover on them and most supermarket produce sections don't really do that much business in hot peppers, so they tend to sit for a while. My favorite place to buy green chilies is at Indian markets (the busier, the better for turnover), second favorite is other Asian food markets, then other specialty markets from places that like spicy food, then farmers' markets, then a long way down the list, regular grocery stores.
When I get them home, I clean them off by hand (remove any stray leaves or any bad ones I accidentally missed, etc) and put them in the fridge in a lightly closed baggie. If I bought a large amount (or if I happen to be growing them and have a lot ripe at once), I will sometimes freeze them. I have a small bag of frozen habaneros at home right now. The texture is most definitely affected by freezing, but the flavor isn't (unless you keep them so long that they get freezer-burned), so if you are planning to cook with them, freezing is perfectly fine.
I usually don't wash them before storing, just pick them over. I wash them just before using them, whether I get them from the garden, the store, the fridge or the freezer.
If I have chilies I'm not sure about, I do smell and taste them to see if they are ok. If they feel/smell/taste weird, I don't put them in the dish. I'd rather replace green chilies with red pepper powder instead of using something iffy.