Proper drawing material specification for steel
Like all things, that depends...
if the strength or heat treat or cold work history is unimportant (rare) then merely calling out the alloy number either via UNS number or AISI alloy number is fine. but usually you need to specify the strength or process history required for the application. do you need it cold worked? heat treated? Is there specific hardness limits required or strength limits required?
I prefer to use an ASTM standard when possible. Usually materials produced to an ASTM standard have controls for both strength and/or hardness as well as chemistry, although that is not always the case. It depends on the specific standard. for instance ASTM A36 does not need a min yeild, neither does ASTM A574 Gr50 because the minimum strength is specified as part of the standard. occasionally for ASTM A500 Grade B/C you'll need to specify a min yeild because grade C is relatively new and rare compared to grade B you may find you have to sort through grade B material certs to find a lot that is as strong as grade C but is not specified as grade C.
Barring the availability of a specific ASTM standard that meet's your needs, you'll have to get more verbose and specify the heat treatment/max or min hardness/min yeild/minimum or maximum cold work level required for the application. but this can get to be a lot of text.
One methodology I've seen used to good stead is if your company deals with only a few materials, is to generate a set of company specific standard materials documents for your company, where the standards document carries all the specific information including all the UT/NDE/ mechanical properties/ heat treating/cold work/ chemistry/ and industry standards it must meet. And then just call out that company standard on the face of the drawing. The down side is that now you have to issue two documents instead of one when you bid out the job etc. and the standard has to be a controlled document and you have to remember that if you change it later, you'll essentially be changing a bunch of other documents indirectly as well so instead of changing the standard, it may be better depending on the circumstances to just generate a new standard.