tl;dr - If you want code compliance, drill holes in the rafters* to run the wiring through that (remember to drill holes in appropriate sizes and location in the rafters), or use a running board.
When running the wiring across the face of the rafters, you need a running board per the NEC, or you'll need to chase the wiring through holes drilled into the rafters. Many inspectors will approve wiring across rafters or joists without a running board (or just miss it), especially if inspecting an attic, since that requirement is typically for basements where someone might want to hang something like wet clothing from it, but it only takes an inspector requiring it one time when you haven't done it to screw you over and force you to redo it to code anyway.
Note: There's an allowance for running NM wiring unsupported if it's 7ft or higher above the floor surface (since it's less likely to snag something on it or hang something from it if it is that high, I suppose), but unfortunately from your photo it looks like your attic is not that tall, so that exception wouldn't apply.
Here's what I believe the relevant NEC is if you're working with NM cable (all excerpts from 2020 NEC):
334.15 Exposed Work:
In exposed work, except as provided in 300.11(B), cable shall be installed as specified in 334.15(A) through (C).
(A) To Follow Surface
Cable shall closely follow the surface of the building finish or of
running boards.
(B) Protection From Physical Damage
Cable shall be protected from physical damage where necessary by rigid
metal conduit, intermediate metal conduit, electrical metallic tubing,
Schedule 80 PVC conduit, Type RTRC marked with the suffix -XW, or
other approved means. Where passing through a floor, the cable shall
be enclosed in rigid metal conduit, intermediate metal conduit,
electrical metallic tubing, Schedule 80 PVC conduit, Type RTRC marked
with the suffix -XW, or other approved means extending at least 150 mm
(6 in.) above the floor. Type NMC cable installed in shallow chases or
grooves in masonry, concrete, or adobe shall be protected in
accordance with the requirements in 300.4(F) and covered with plaster,
adobe, or similar finish.
(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 (1/4 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.
334.23 In Accessible Attics
The installation of cable in accessible attics or roof spaces shall also comply with 320.23.
Which is:
320.23 In Accessible Attics:
Type AC cables in accessible attics or roof spaces shall be installed as specified in 320.23(A) and (B).
(A) Cables Run Across the Top of Floor Joists
Where run across the top of floor joists, or within 2.1 m (7 ft) of
the floor or floor joists across the face of rafters or studding, the
cable shall be protected by guard strips that are at least as high as
the cable. Where this space is not accessible by permanently installed
stairs or ladders, protection shall only be required within 1.8 m (6
ft) of the nearest edge of the scuttle hole or attic entrance.
(B) Cable Installed Parallel to Framing Members
Where the cable is installed parallel to the sides of rafters, studs,
or ceiling or floor joists, neither guard strips nor running boards
shall be required, and the installation shall also comply with
300.4(D).
And
334.30 Securing and Supporting
Nonmetallic-sheathed cable shall be supported and secured by staples, cable ties listed and identified for securement and support, or straps, hangers, or similar fittings designed and installed so as not to damage the cable, at intervals not exceeding 1.4 m (41/2 ft) and within 300 mm (12 in.) of every cable entry into enclosures such as outlet boxes, junction boxes, cabinets, or fittings. The cable length between the cable entry and the closest cable support shall not exceed 450 mm (18 in.). Flat cables shall not be stapled on edge.
Sections of cable protected from physical damage by raceway shall not be required to be secured within the raceway.
(A) Horizontal Runs Through Holes and Notches
In other than vertical runs, cables installed in accordance with 300.4 shall be considered to be supported and secured where such support does not exceed 1.4-m (41/2-ft) intervals and the
nonmetallic-sheathed cable is securely fastened in place by an
approved means within 300 mm (12 in.) of each box, cabinet, conduit
body, or other nonmetallic-sheathed cable termination. Informational
Note: See 314.17(B)(1) for support where nonmetallic boxes are used.
(B) Unsupported Cables
Nonmetallic-sheathed cable shall be permitted to be unsupported where the cable:
- Is fished between access points through concealed spaces in finished buildings or structures and supporting is impracticable.
- Is not more than 1.4 m (41/2 ft) from the last point of cable support to the point of connection to a luminaire or other piece of
electrical equipment and the cable and point of connection are within
an accessible ceiling in one-, two-, or multifamily dwellings.
So you need to comply with 334.15. You also need to comply with 334.23, which says 320.23 applies to NM cables even though 320.23 only mentions AC cables. And 334.30 covers the general support requirements.
The entire attic will be a conditioned space (it's not currently, but that's changing very soon)
Okay, since you're air conditioning the space, that means you'll be putting some kind of insulation between the rafters, most likely. If you also put up drywall or some kind of board over the rafters (to create a "finished space"), then none of the above applies, because it wouldn't be "exposed work" anymore.
Of course, for wiring behind drywall, you typically drill a hole through the framing member (in an approved manner, of course), which is also an acceptable alternative to a running board for exposed work, too. If you ran the wiring across the edges of the rafters, then if you ever did want to put up drywall or some kind of 'ceiling' board, you'd have to provide additional protection for the wiring that's in the way (e.g. furring strips with metal plates over the gaps created for the wiring) or re-run the wiring through holes drilled into the joists.
So either way my recommendation is to drill holes, don't run wiring across the face. Future you will thank past you.
* - for any readers who have trusses instead of rafters, you typically are not allowed to drill holes through trusses. There are specific rules/restrictions for that which are not covered here.
