Lexical class of "walking" in this sentence

According to traditional grammar, this is a gerund, indicating the action or process of walking, as in catching butterflies is fun. But there is a catch. Gerunds can be considered a type of noun; as such, it is here used as an adjective to modify trail, just as most nouns in English can be used as adjectives, as in a phone booth, a cookie jar. (It could also be argued that walking-trail has since become a fixed phrase and therefore a noun of its own, but that doesn't really matter.)

Edit: Why is it not a participle? Well, if you have [present participle] [noun], and you were to convert this into a relative clause, like [noun] [relative pronoun] [finite verb], the relative pronoun would be the subject of the verb, and so the antecedent would be the entity that "does" the verb.

Examples: a moving car ⇒ a car that moves; a walking child ⇒ a child that walks. This does not work in the case of the trail: a walking trail ⇒ *a trail that walks. In our sentence, walking is not a participle, because walking is only semantically linked to trail in a general way, not as a converted subject.

In this case, walking is an adjective. From MW:

walking: adj. 2a: used for or in walking <walking shoes>

A noun can be modified by an adjective or by another noun and this remains true when the modifying word ends, as in the example, in -ing. The 'Longman Student Grammar of Spoken and Written English' has an elegant test for deciding which it is. If the clause can be paraphrased using a relative clause, then the -ing form is an adjective. If it can be paraphrased using a prepositional phrase, then the -ing form is a noun. This enables us to distinguish between the adjective running in running water (water that runs) and the noun running in running shoes (shoes for running).

If we apply the test to the example, it becomes clear that we cannot say Pets prohibited on trail which walks, so walking can’t be an adjective. Pets prohibited on trail for walking, however, makes sense, so walking must be a noun.