Electrical across studs in basement?
Yes, there are codes against running nonmetallic sheathed cable (Type NM) along the edge of studs. The cable must be protected from physical damage (NEC 300.4), which it would not be on the edge of the studs.  

If you don't want to bore holes, you can use schedule 80 PVC conduit (or any conduit rated to protect cable).

Even if you do bore holes. Since you're not covering the wall, the cable could still be considered "exposed".  In which case, it would need to be protected from physical damage.