Hi Paul,

As explained in the following lesson -

https://kwiziq.learnfrenchwithalexa.com/revision/grammar/monter-can-be-used-with-avoir-or-etre-in-le-passe-compose-and-changes-meaning

it depends whether the verb 'monter' has an object (transitive) or not .

In this case it has - monter les escaliers - so it will take 'avoir' but if you said 

Je suis monté dans ma chambre = I went up to my bedroom 

it would be 'être' as it has not object (intransitive).

Hope this helps!