Responsibility for the stairs can be shared
Common stairs and landings are usually the owner’s to maintain. Whether a guest injured on them has a claim against the owner, against you, or against both depends on the circumstances and on what you knew. Your liability cover is what answers your share, and the limit is usually whatever the lease demanded.
Egress is a safety question before it is a legal one
A single exit, a blocked back stair or a window that will not open is worth raising in writing with the owner immediately. It is a habitability matter, and documenting that you raised it protects you in more ways than one.
Working smoke alarms are both parties’ interest
Owners generally must provide them and tenants should not disable them. Beyond the legal position, in an old timber-framed building the alarm is the thing that gets you down the stairs. It is worth testing rather than assuming.