Scendere can be both intransitive and transitive. The transitive meanings are:
- Going down something: this is the case you found in the book: scendere le scale = going down the stairs
- Bringing down something: used in some local dialects: «Scendi lo zaino» = «Bring down the backpack».
As a reference, Treccani online dictionary has for scendere⁴:
4. tr. a. Percorrere in discesa: sc. le scale, un’erta, un pendìo; fece un salto, scese il letto in furia (Manzoni). b. region. Calare, spostare o portare in basso: rinvoltammo il padrone dentro una pelliccia e delle coperte, lo scendemmo e caricammo nella macchina (Palazzeschi); l’avevamo sceso [il carro] dal crocevia e sistemato molto bene tra i cespugli (Arpino).
Transitive verbs in Italian require mostly the auxiliary avere, and scendere makes no exception.
A number of related questions on the internet, as well as this very question and my previous erroneous answer, show that the transitive form of scendere is perceived as wrong by some native speakers. This is probably an instance of hypercorrection, due to the transitive form being used often in its second meaning in southern dialects (see also l'ipercorrettismo e i dialetti).
An article by Accademia della Crusca about the transitive forms of entrare, uscire, salire, scendere, while being concerned mainly with new and dialectal forms, states in the second-last paragraph that transitive forms of scendere are correct: «[Salire e scendere] ammettono anche in lingua la possibilità di un uso transitivo e quindi [...] richiedono l’ausiliare avere: sono salito da te e sono sceso all'alba, ma anche ho salito la gradinata e ho sceso le scale».
Also, an article on the Treccani website answering this same question states that the transitive form is correct and its meaning can be different from that of the intransitive form «sono sceso dalle scale», as pointed out by @DaG in a comment.
Historical usage

The graph from google ngrams shows occurences dating back to 1808, but they come from even older texts, like Galileo Galilei's Dialogo sopra i due massimi sistemi del mondo (1632): «[due pietre] fusser talmente differenti nelle loro velocità, che [...] l'altra si trovasse non avere né anco sceso dieci braccia»
Use of the passato prossimo is more recent, yet old enough to be accepted as correct. This is probably due to this tense becoming increasingly common through the last century. The form is also used by many classical authors, as pointed out by @DaG. An example by Eugenio Montale: «Ho sceso, dandoti il braccio, almeno un milione di scale».