It is poor English, so it requires some interpretation. I think they were going for: "Corporate drones are quick to engage their damage containment routine." As Natalia said, a corporate drone is a mindless robot for the corporation. This continues the analogy by saying that they apply their programming ("routine" = program). "Engage" makes it sound more mechanical and basically means "start" or "apply" here. "to do" just sounds like the wrong verb to me. "damage containment" might be a somewhat poor attempt to again make it sound more robotic / mechanical by using slighting the wrong word but keeping the right overall meaning. As Doctor Rod said, "damage control" would be the normal phrase.

"Corporate drones are quick to do damage containment routine." First of all, this is a terrible English sentence. Nobody does "damage containment." We talk about doing "damage control." Also, "routine" is a noun and does not properly say anything about the containment or the "damage containment." I'm not sure what it means! Maybe the author of the sentence is trying to say that a corporate drone quickly works to contain damage on a routine basis, or "routinely." Here is a good English version of this sentence, assuming I understand the author's intent: "Corporate drones are routinely quick to do damage control." This means that mindless employees of corporations [drones] quickly act to control damage to the company [to its reputation or finances] on a regular basis [routinely].

CORPORATE DRONE (n.) A mindless robot for the corporation. Someone who has "sold out". Goes along with all management initiatives and enjoys repeating corporate "jokes" like "is it Friday yet", and "didn't you get the memo about the red shirt" And they damage something quickly))) Aren't there any commas or prepositions missing?

Oh, that's it) they do routine (some ordinary repeated actions), not damage or containment :) What sort of routine? - containing (in the meaning of controlling, supressing, restricting) damages Thanks Scott and Doctor Rod