Making negative mass won't be easy. If you go down the convert energy into mass it will be the same energy requirements for making antimatter. In this case, just look up information on antimatter production.
Now if you're in the business of using negative for "hypothetical warp-drive technology", then if we make some pretty hairy assumptions it becomes plausible in a science-fictional way.
Spacetime is seething, down at the microcosm, with virtual particles which pop into and out of existence in astronomically small amounts of time. This is mass energy borrowed from the quantum vacuum. Quantum theory suggests, and this is where it starts getting very hairy, that the quantum vacuum has a higher mass energy density than nuclear matter.
Let's assume that this quantum vacuum consists of nearly equal amounts of negative and positive mass energy. We shall also assume that there exists a technology that pull equal amounts of both forms of mass energy out of the quantum vacuum. We will assume once out of the quantum vacuum this mass energy can be sustained indefinitely. Actually this assumption isn't entirely necessary as for some technological applications all you would need to do is continually maintain the equal amounts of mass energy. For example, in a diametrical drive system. For more information go here and here.
The hypothetical quantum vacuum extraction device will need energy to run. If you're doing this for a science-fiction story, you can choose how much is needed for the machinery. On the safe side, assume it's quite large.
Negative mass can be used for diametrical drive spacecraft which should be capable, in theory, of accelerating to near-lightspeed, and for some forms of antigravity. For your "hypothetical warp-drive technology" and being able to keep wormholes open you so-called exotic matter. This is much stronger stuff and while I can see a purely hypothetical way of obtaining negative mass I haven't figured a suitable way to conjure up exotic matter.
Note: this answer refers to mass energy because covers both matter and energy and I've used that as shorthand instead saying matter and energy all the time.