Sometimes it starts with simply mastering adjectives. An English teacher in high school would not permit the use of "fine" or "okay" as an answer. Perhaps a simple start with a person using a more descriptive word like moody, overwhelmed, excited, nervous etc, leaves more invitation for further discussion. Then, simply follow with a statement if you want to go no further, or leave it open for the other person to inquire if they want to listen and help.

If you feel like offering help but people are not asking for it, simply say that you were wondering if they could use your help with anything at the moment, or ask how their day is going and let it be known you have an ear to bend. Beware though of people who just want to waste your time.