Is it okay to staple romex to a floor joist when running it perpendicular behind a finished basement ceiling?
You can run cables across and under the joists. If you do though you must install them on running boards for protection. Typically it is just easier to drill.

National Electrical Code 2011
Chapter 3 Wiring Methods and Materials
Article 334 Nonmetallic-Sheathed Cable: Types NM, NMC, and NMS
II. Installation
334.15 Exposed Work.
(C) In Unfinished Basements and Crawl Spaces. Where cable is run at angles with joists in unfinished basements and
crawl spaces, it shall be permissible to secure cables not smaller
than two 6 AWG or three 8 AWG conductors directly to the lower edges
of the joists. Smaller cables shall be run either through bored holes
in joists or on running boards. Nonmetallic-sheathed cable installed
on the wall of an unfinished basement shall be permitted to be
installed in a listed conduit or tubing or shall be protected in
accordance with 300.4. Conduit or tubing shall be provided with a
suitable insulating bushing or adapter at the point the cable enters
the raceway. The sheath of the nonmetallic-sheathed cable shall extend
through the conduit or tubing and into the outlet or device box not
less than 6 mm (¼ in.). The cable shall be secured within 300 mm (12
in.) of the point where the cable enters the conduit or tubing. Metal
conduit, tubing, and metal outlet boxes shall be connected to an
equipment grounding conductor complying with the provisions of 250.86
and 250.148.