How should I run electrical wire perpendicular to exposed joists in my basement?
Conduit is not required where a cable runs through bored holes in joists. If you're worried about it, you could install running boards between the joists above or below the cable.

National Electrical Code
Chapter 3 Wiring Methods and Materials
Article 334 Nonmetallic Sheathed Cable: Types NM, NMC, and NMS
334.17 Through or Parallel to Framing Members. Types
NM, NMC, or NMS cable shall be protected in accordance
with 300.4 where installed through or parallel to framing
members. Grommets used as required in 300.4(B)(1)
shall remain in place and be listed for the purpose of
cable protection.

Fire retardant is also not typically required, unless the cable is passing between dwelling units, into a garage, or between floors.

National Electrical Code 2014
Chapter 3 Wiring Methods and Materials
Article 300 Wiring Methods
300.21 Spread of Fire or Products of Combustion. Electrical
installations in hollow spaces, vertical shafts, and
ventilation or air-handling ducts shall be made so that the
possible spread of fire or products of combustion will not
be substantially increased. Openings around electrical penetrations
through fire-resistant-rated walls, partitions,
floors, or ceilings shall be firestopped using approved methods
to maintain the fire resistance rating.

This is of course unless there are any local amendments.