Is there any way to quickly check if the games you have on steam can drop cards?

Enhanced Steam will show you how many games have waiting drops.

Also another option would be to use a card idling program like asf or idlemaster and find out what games had drops through there.

I didn't understand what you were asking, so this won't help. I'm not sure iif Enhanced steam will but I'd suggest trying that.

Why do you have to install the game to find out if you have card drops? Aren't they listed in you badges with card drops remaining? Or do those disappear if you don't get the drops after a while?

If you go to your badge page, this will show all the card drops still available for your games...just install the ones you want to card farm from

Several ways to check what you've asked in the thread topic, but judging by your comment above you want to also see all games you've already farmed for cards (and traded/sold away)?

In that case, there's only one approach I'm aware of:
Go to your Badges page [Username>Community>Badges] and click 'View my booster pack eligibility' (right above the sorting methods in the top right).
You're going to have to sift through it manually to figure out which ones you haven't done, but it does show the complete list of games which have badges (except those you haven't farmed for cards yet).