The choice of dative or accusative after two-way prepositions has nothing to do with the verb as such.
It's to do with whether you are describing a location, or the destination of motion.
Here, there is motion; but the walking takes place only on the street. The street is this the location of that motion, not the destination or endpoint or target of motion which starts somewhere else.
There's movement involved. Shouldn't it be in akk?
The accusative is used when the noun is the destination of motion.
- Er läuft auf die Straße. = He runs onto the street. (e.g. from a house to the street)
- Er läuft auf der Straße. = He runs on the street. (The street is the location of the running but not the destination.)