Can we influence the way people remember us?

Yes, most of people's memories are based on olfactory senses (taste, touch, and smell). If you wear a certain perfume then it can trigger their memory of you.

Now if you mean how they remember you as a person, it depends on how significant you are in their life. There are many people in my life that I've met but they never 'impacted' my life in any way. Anyone who I have fought with or been best friends with and loved I remember because they were prevalent in my life.

The way our brain works is by storing the memories of a person in the form of short clips or events that have taken place. The amygdala is the brain region that stores the experiences in the form of feelings. When we meet someone with whom we have had good memories, our amygdala tells us that it is safe to be with them, and makes us feel positive about them. The behaviour of the person in the front influences how we perceive their actions and needs. This affects the memory formation, as our brain may call them noise and not store the person's identity.