Most people won't approach for no reason at all. Below you'll find some reasons that people will approach characters, as well as reasons why characters may notice someone. Let this serve as a springboard for other ideas.
1- Recognizes a character from someplace.
2- Being chased by the city guard.
3- Just killed someone and doesn't know what to do.
4- Sees something they want from party - will use treachery if necessary.
5- Sees something they want from the party - will pay anything.
6- Being hunted by a monster.
7- Has a disease.
8- Is actually a predator of some sort.
9- Has magical item they want to sell.
10- Has a non-magical item they want to sell.
11- Recommending a restaurant, etc.
12- Wants to hire adventurers.
13- Knows a secret, wants to tell someone.
14- Looking for companionship/friendship.
15- Complaining.
16- Needs directions.
17- Person is fantastically beautiful or thinks a character is beautiful.
18- Person looks familiar.
19- Wearing nearly identical clothes to a character's.
20- Character remembers seeing the person in a dream.