I sometimes find questions that are normally ended with the question particle か to end with かい. For example:
いいかい？
そうかい？
あいつがどうかしたかい？
And sometimes with just い without か (if I understand this one right):
なんだい？
Is かい a softer version of か? How is it used?
This usage of 「かい」, in real life, is largely limited to male speakers of the Kanto region and even among those, they only use it with people that they know very well and that are as old as or younger than the speaker. It must also be made clear that it is very informal. Do not ever use it with a stranger or someone you have just met on the street.
Regarding the softness, 「いいかい？」, for instance, surely sounds MUCH softer than 「いいか?」. In fact, 「いいか？」 sounds pretty curt or condescending -- perhaps much curter than many Japanese-learners would imagine if I may speak from my experience with J-learners.
「だい」 as in 「なんだい？ = "What is it?"」 is also largely Kanto-masculine-friendly.
Since both 「かい」 and 「だい」 have a couple of widely different usages, I am being careful not to mention (and confuse people) by discussing the usages other than as a casual question-ender. In fact, @Chocolate left a comment ending with a mostly-Kansai usage of 「かい」 as a joke but kindly removed it lest it might confuse the learners. S/he used the "urging かい" to say 「自分{じぶん}で答{こた}えんか～い」= "Why don't you answer the question yourself?"
Yes, かい is softer than just か, but can also denote a yes-or-no question.
Example:
狂ったのかい？ Are you mad?
This is a yes-or-no question. I suppose that you could answer in a way other than yes or no, but it seeks a yes or a no. It's also softer, possibly to placate the person who is mad.
Hope this helps.
