'On foot' is the more commonly used expression.This also stems from the fact that 'on' is usually used for actions involving body parts.
'By' is usually used to talk about a means of transport (i.e. train, car, boat, plane, etc).
They both have the same meaning, albeit with different grammatical structure.
We can use say both – with these justifications:
- Why you should say on foot
We usually use on for movements or actions that involve using body parts. You can rest on your elbows, you can pray on your knees, and you can lie on your back. Walking is no different – the action takes place on foot.
- Why you could say by foot
We use by to describe a tool used for movements or actions. You can travel by plane, by boat or by car, for instance. You can send letters by post, and you can write by hand. It stands to reason, then, that you can travel by foot.
- So which is more correct?
You move with your feet in contact with the ground, supported by your feet, making on foot a more literal description of the action. Yet your feet are also a tool of sorts, so if we want to describe what you use to travel, by foot makes perfect sense. It is not reasonable, then, to say one is more correct than the other. Yet you may be told that by foot is incorrect. Why? For the same reason that many rules exist in English:
Because ‘on foot’ is more commonly used than by foot.
Source: English Lessons Brighton
The following links also explore the problem in depth:
“By foot” vs. “on foot” (EL&U)
http://www.icaltefl.com/on-foot-vs-by-foot